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Methods for evaluation of elliptical profiles
Int. J. Mach. Tool Des. Res. Printed in Great Britain
Vol. 25, No. 4. pp. 299--312, 1985.
0020-7357/8553.00+.00 Pergamon Press Ltd.
METHODS FOR EVALUATION OF ELLIPTICAL PROFILES T. S. R. MURTHY* (Received 6 September 1982; in final form 5 December 1984)
Abstract--There are many interesting properties of elliptical profiles used in precision instruments which simplify the mechanisms to obtain a desired function. One such application is a spectrophotometer developed at CMTI. The other applications are elliptical gear blanks, pistons of I.C. engines etc., whose cross sections are elliptical. Elliptical sections having very little difference between major and minor dia. can be measured with the roundness measuring instruments to obtain the difference between the major and minor dia. However, fitting of best fit ellipse and hence the definition of eilipticity are not available. In this paper 3 different methods are proposed for the evaluation of the elliptical profiles, based on normal least squares fit, bivariate Gaussian distribution and general second degree equation. In the 3rd method the deviations from the property of the ellipse are minimised.
NOMENCLATURE A B D E,, E~
N ai a
b C
dr dk eA,eB
f
h ml,m2
P i ,P2 r I ,r2
b/ v
x',y' xi,Yi xp,yp
0 P ~x ~v ~xy
+(x,y)
constant (coefficient of x 2) in the general second degree equation in x and y constant (coefficient of y2) in the general second degree equation in x and y constant in the general second degree equation in x and y sum of the squares of the normal deviation sum of the squares of the deviation from the property of the ellipse number of approximate equi-angular spaced data points constants in the general second degree equation in x and y X-coordinate of the centre of the best fit ellipse Y-coordinate of the centre of the best fit ellipse constant in the equation of a straight line y = mx + c distance from a point (xk,Yk) to a point on the ellipse normal distance from measurement point k on to the best fit ellipse ellipticity values constant (coefficient of xy) in the general second degree equation in x and y difference in the semi-major and semi-minor dia. of two concentric ellipses slope of the major axis and slope of the minor axis of the best fit ellipse semi-major and semi-minor dia. of an ellipse (representing constant probability) semi-major and semi-minor dia. of the best fit ellipse constant (coefficient of x) in the general second degree equation in x and y constant (coefficient of y) in the general second degree equation in x and y local coordinate axes passing through the centre and along the major axis and minor axis of the best fit ellipse x and y coordinates of the ith measured point on the ellipse x and y coordinates of a point on the curve parallel to the ellipse orientation of the major axis, tan a = m angle made by the normal (to the ellipse at a point on it) with the x-axis polar coordinate correlation coefficient standard deviation of x standard deviation of y co-variance (x, y) joint probability distribution INTRODUCTION
MEASUREMENTand evaluation of circular profiles of cylindrical and spherical components in terms of circularity [1], cylindricity [2-4] and sphericity [5] has been well established by the use of roundness measuring instruments. Elliptical sections having very little difference between major and minor dia. can be measured with roundness measuring instruments to obtain the difference between their dia. However methods for fitting the best fit ellipse and definition of ellipticity (form deviation of elliptical profile) are not available. * Central Machine Tool Institute. Bangalore 560 022, India. 299
300
T . S . R . MURTHY
There are many interesting and useful applications for elliptical profiles in engineering practice used in instruments like a spectrophotometer developed at CMTI for obtaining various incidence angles between 2° and 60°. The other applications are pistons of IC engines (a special manufacturing technology developed at CMTI [9]) and elliptical gear blanks. Since no method for evaluation of elliptical profiles are available an attempt has been made in this paper to define its form deviation namely ellipticity and devise methods for its evaluation. Three methods for the construction of the best fit ellipse are proposed. The first method is called a Normal Least Squares Method which simultaneously fits the major and minor axis to the data corresponding to ellipse. In the second method the measured x and y coordinates of the elliptical profile are assumed to follow bivariate Gaussian distribution and hence the parameters of this distribution are related to the constants in the equation of the ellipse namely the centre of the ellipse and the orientation of the major axis. The third method fits a general second degree equation in x and y to the data by minimizing the sum of the squares of the deviation from the property of the ellipse. These methods are directed towards the ultimate objective of developing efficient algorithms for computer aided metrology and manufacture. ELLIPSE AND ITS PROPERTIES
Ellipse can be defined and understood in different ways. Appendix A shows some of the definitions, important properties of the ellipse and their engineering application. The standard canonical form of the ellipse is given by x 2/ ~ ") + y 2 / ~ =
1.
DEFINITIONS
The term ellipticity is generally used to mean the deviation from the form of a circle or sphere. The coefficient of compression of ellipse (Appendix A) is rCr2, its reciprocal is called axial ratio and ellipticity in mathematics is defined as 1-r2/r~. But for engineering metrology application the term ellipticity has been defined similar to circularity to represent the deviation from the form of an ellipse. For defining ellipticity construction of inner and outer parallel curves to ellipse is helpful. In the polar form the equation of the ellipse shown in Fig. 1 is given by x = rt cos 0 y = r2 sin O.
(1)
If 13is the angle made by the normal at a point on the ellipse the locus of the point with constant distance h from the ellipse is given by xp = rl cos O + h cos~3
yp = r2 sin 0 + h sin 13
(2)
where 13= tan-~ (rl tan O/r2) is the angle made by the normal to the ellipse at any point on it. (Tan 13 = - d x / d y is obtained by differentiating equation (1).) Equation (2) is not an ellipse with semi-major and minor dia. increased by h. If the semi-major and minor dia. are increased by h the corresponding equation of the ellipse would be x = (rl + h) cos0 y = (r2 + h) sin 0.
(3)
Figure 2 shows the plot of equations (1), (2) and (3) with h positive for outer curve and h negative for inner curve to the ellipse given by equation (1). From Fig. 2 the
M e t h o d s for E v a l u a t i o n of E l l i p t i c a l Profiles Locus o f a
t Y
-- -- -- -- -- /
301
paraLLeL curve x p, y . )
h
\
\
o rl cos 8
;I
I-
FI6. 1. C e n t r a l ellipse and the locus of a p o i n t with c o n s t a n t n o r m a l d i s t a n c e , h.
~Parattet \equation
curve "to eLLipse ( 2 ) with negative
__
tY ~
,
L_ equation ( 2 ) with 'h' positive
ti
x2
EL pse i7~~_h~2 ,-i--,
+
y2
,._;-'SZT2 ,,~,--,
....
tr~n~
¢t/n~=
o x~
o Y" -
,,..++ -'~-~ + -~-z~ - I x
~r tn+
F1G. 2. E l l i p s e and d i f f e r e n t p a r a l l e l c u r v e s to ellipse.
following points should be noted: (a) Equation (2), namely parallel curve to the ellipse, and equation (3), namely an ellipse with equal increase in semi-major and minor dia., are different. Both of them meet at 4 points i.e., at the extremities of major and minor dia. when outer curve is considered. The curve given by equation (2) is continuous when h is positive. (b) With h negative a discontinuous inner curve is obtained with fish tail shape at the extremities of the major axis. (c) The area enclosed between the inner and outer curves given by equation (2) is larger compared to the area enclosed between the inner and outer ellipses given by equation (3) with same h. Because of the above difference between equations (2) and (3) two definitions for ellipticity are brought out: Ellipticity eA: is defined as the scaled difference along the semi-major axis of two concentric ellipses constructed around one of the specified best fit ellipses which are just sufficient to contain the total elliptical profile. Ellipticity eB: is defined as the scaled difference along the semi-major axis of two concentric curves constructed parallel to one of the best fit ellipses which are just
302
T.S.R.
MURTHY
sufficient to contain the total elliptial profile (Fig. 2). The centre of the best fit ellipse and the orientation of the major and minor axes are constructed so as to satisfy any one of the following: (a) Minimum Zone (separation) Ellipse (MZE) is the mean ellipse which gives ellipticity eA or ea minimum. (b) Least Squares Ellipse (LSE) is the mean ellipse satisfying the least squares condition for the property of the ellipse. (c) Maximum Inscribed Ellipse (MIE) is the largest ellipse that can be inscribed within the elliptical profile. (d) Minimum Circumscribed Ellipse (MCE) is the smallest ellipse that contains the elliptical profile. The minimum zone ellipse will be the preferred best fit ellipse because it gives least error. Since MZE has to be found by search techniques, [6] requires a starting solution. Hence methods for finding the centre and orientation of the best fit ellipse have been dealt with in this paper. METHODS OF EVALUATION
When the measurements are taken at specified points on the ellipse, then the ellipse in engineering terms is represented by the measured points itself in rectangular or polar coordinates. As the coordinate system can be easily interchanged it is assumed that the coordinates of measured points on the ellipse are available for analysis. (a)
M e t h o d o f n o r m a l least squares
This method determines the centre of the ellipse and the orientation of major and minor axes. Considering measured points xi, Yi to be available (Fig. 3) on the elliptical profile it is possible to fit for the measured data a general least squares straight line of the form y = mx + c to represent the major axis by minimizing the sum of the squares of the deviations (~,PM2 = Z ( y i - y ) 2 ) . Similarly it is possible to fit a general least squares line of the form x = m ' y + c' to represent the minor axis. If these two lines are mutually perpendicular then they can be taken to represent the major and minor axes. This means ram' = - 1.
~ J
•ly:m2 JMinoraxis x +%
\
Majoraxis
\ Y 23 ~
0
FIG.3.
I
I
I
y~)
N
I
I
2
3
/1
~rla'b)
I
4 X
~
~ll~ 5
2: 190+ a~). I I 6
7
Normal least squares method for fitting the axis of the ellipse.
Methods for Evaluation of Elliptical Profiles
303
But in the general least squares methods the product m m ' i s not exactly - 1 , but m m ' = r~ = correlation coefficient which lies between 0 and + 1. A method which automatically satisfies the condition that the product of the slopes of the straight lines representing the major axis and minor axis is - 1 (ram' = - 1 ) is called normal least squares method by the author and has been first used in finding the minimum zone evaluation of surfaces [6] and extended now to determine the orientation of the axes of the elliptical profiles. In the normal least squares method the sum of the squares of the perpendicular distances (Fig. 3) from a point P (xi, Yi) on to the straight line of the form y = m x + c are minimized. (4)
En = £ P N z = £[(Yi-Y) cos ot] 2.
It is shown below how two mutually perpendicular straight lines are fitted by normal least squares method from the above equation. Substituting y = taxi + c and cos 2 t x = l / ( l + m 2) in equation (4) gives
(s)
E,, = Y ~ ( y i - m x i - c ) 2 / ( l +m2).
Differentiating partially equation (5) with respect to m and c and equating to zero the equations to solve for m and c are
{[N~x 2 -- (Y, Xi) 21 -- [NZy 2 - (£yi)2] } m2 + m
[N'ZxiYi
~ j
and c = Y. ( y i - m x i ) / N .
- 1 = 0
(6) (7)
Equation (6) is quadratic in m (and has two solutions ml, m2) and the constant term is - 1 (product of the roots m m2 = - 1 ) . Hence equation (6) gives the slopes of the two mutually perpendicular straight lines fitted by normal least squares method. The intersection of these two mutually perpendicular straight lines gives the centre (a,b) of the ellipse as derived below. If these two straight lines are given by y = m ~ x + c l and y = m 2 x + c 2 ; substituting a for x and b for y and solving for a and b a = (c 1 - c 2 ) / ( m
l+m2)
substituting cl, c2, ml, m2 from equations (6) and (7) a = Exi/N
and
b = Y.yi/N.
(8) (9)
Since a and b are average locations along x and y axes the solution for a, b, mr, m2 are suitable for the construction of the best fit ellipse. (b)
M e t h o d o f joint probability distribution
This method is applied here mainly to determine the centre (a,b) of the ellipse and the orientation of the axes. In this method the property of the joint normal distribution of two variables x and y (bivariate Gaussian distribution) which shows that the curves of constant probability are ellipses is made use of to determine the centre and orientation of the ellipse.
~i~oR 2 5 : 4 - B
304
T.S.R.
MUllTH¥
The joint normal distribution (Fig. 4) of the two variables x and y is given by [7]
O0(x,y) =
2~rex~y X / ( l - p 2) exp
x-a
.l:xa:ox,
[
y-b
2(1_p2 )
{ y-b
~2l]
) ( W. ) + ," "Ho--V,
where p = cov (x,y)/(cr,~ry) = Cr2x/(~rxCry).
(10)
(11)
In equation (10) the curves of constant probability are determined by equating the exponent to a constant K. i.e.
x--a
, ox
(y-b
)
(y-b] 2
+ ,o,,--
--
This is the equation of an ellipse with centre (a,b) as shown in Fig. 5 with parameters ~.~, ~v and 9. The principal axis of the ellipse makes an angle a with respect to the x-axis as shown in Figs 3 and 5. The angle e~ and the semi dia. Pt and P2 are derived from the well known properties of conic sections as tan 2e~ =
p2 =
2 p ~ x O r y / ( C r x2 --ffy) 2
(13)
2 2 2 2 ot--29CrxCrysin a cos a + Crx 2 sin2tx) crxcrrK(19 2 )/(CryCOS
p2 = cr~o~yK(a_p2)/(crZsinZa+2pCrx%,
sin a cos a + ~ cosZoO.
(14) (15)
Since we are interested in the centre (a,b) of the ellipse given by equation (12) and orientation a, their maximum likelihood estimates are given by
a = Exi/N = £ b = Eyi/N = 37 O"x2
y~(xi_a)2/N
2 O'y
= Z(yi_b)2/N p = X(xi-a) ( y i - b ) / ( N c r x ~ ) a n d 2 "~ tan 2a = 2pCrxCry/(Crx-~7.).
(16) (17) (18) (19) (20) (21)
Determination of major and minor d&. After locating the centre and orientation of the ellipse by the above methods semi-major and minor dia. have to be determined to construct the best fit ellipse. This requires a least squares criterion which is slightly different from the general least squares method and is described below. For convenience and simplicity the coordinates of the data points (xi, Yi) a r e measured with respect to a new coordinate system (x',y') with the origin at (a,b) and x-axis along the major axis so that the equation of the ellipse now is of the form x,2/~ + y,Z/~ = 1.
(22)
If a point (x[, y[) lies on the ellipse then equation (22) will be exactly satisfied, i.e., left hand side term will be exactly unity. If it is not on the ellipse the left hand side term will not be exactly unity. The sum of the squares of the deviation from unity has been minimized for the solution. These deviations are named as the deviations from the property of the ellipse and are given by
Es = Z(x:Z/r 2 + y:2/~ _ 1)2
(23)
Methods for Evaluation of Elliptical Profiles
305
~(x,y)
FIG. 4. Probability density of a bi-variate Gaussian distribution.
For K=I
Moior a x i s /
3.6 3.5-2.8-2.4-Y 2-1,6-1.2-0.8 -0.4 --
I
0.4
08
I
I
12
I
15
2
I
2.4
I
2.8
I
32
X
FIG. 5. Co-variance ellipse.
= X(Ax; 2 + By 2 - 1) 2 where A = 1/~ and B = 1/~.
(24)
Differentiating equation (23) partially with respect to A and B and equating to zero the equations to determine A and B and hence rl, r2 in matrix form are given by
X//4
Xx;2y;2
A Xy;2 ] .
(25)
(c) Method of best fit general second degree equation in x and y Methods (a) and (b) are a step by step approach to determine first a, b, a and then rl and re. In this method a general second degree equation in x and y of the form given below is assumed.
306
T.S.R.
A'x 2 + B'y 2 +
MURTHY
f x y + u'x + v'y - D = 0.
(26)
If D :k 0, equation (26) is rewritten by dividing by D as Ax 2
+ By 2 + fxy + ux + vy - 1 = 0
(27)
where A = A ' / D etc. T h e sum of the squares of the deviation from the p r o p e r t y of the ellipse (conic section) is given by E, = (Ax 2 + By2i
+
fxiYi + uxi +
vyi --
1) 2.
(28)
Differentiating partially with respect to A, B, f, u, v and equating to zero the equations in the matrix form to d e t e r m i n e A, B, f, u, v, are given by equation (29).
x~4
Xx~
Xx~, Xx~
x~,
2x~
Exi2y2
Ey~
Exiy 3
Ey~
Yy~
x@,
Xx~v~ x@ 2 Xx~,
Xx,y?
Ex,~¢i
Ex 3
Ex,y2~ Ex~,
Ex~
Ex,yi
Yxi
Xxi2Yi
Xy 3
Exzyi
E,y:i
Zxiy 2
Exiy 2
v
Eyi
(29)
I
J
F r o m A, B, f, u, v the diameters of the ellipse and its centre etc. are obtained as (30) (31) (32)
a = ( v f - 2 B u ) / ( 4 A B - f 2) b = ( u f - 2 A v ) / ( 4 A B - f 2) tan 2~ = f / ( A - B ) - - u v f + ( A v 2 + B u 2) +
4(AB -f2/4)
~' ~ = 23~-B~f~ i ~ + ~ - ' ~ ~
"
(33)
T h e s e quantities are e v a l u a t e d with the aid of invariants of the second o r d e r curve as shown in A p p e n d i x B. COMPARISON
It is interesting to note that there exists some relation among the three methods. In the m e t h o d of n o r m a l least squares using the relations 2
(~ X
Similarly
~
E(Xi
X)2/N
or [NEx 2 - (Exi) 2] = U2cr2. • [N'Zy 2 - (Eyi) 2] = U2cr~, (N~,xiy i - ~,xi~,,yi) = N 20.xy 2
and substituting t h e m in e q u a t i o n (6) m 2 + m [(0.2 - 0.~)hrxy] 2 2 - 1 = 0
(34)
(35) (36) (37)
(38)
using 2
D = (Txy/ITxffy, w e get
m 2 + m [(or2 - trz)/(pcr.,cr:.)] - 1 = 0.
(39)
M e t h o d s for E v a l u a t i o n of E l l i p t i c a l Profiles
In the above equation since mira 2 = - 1 if m I = tan o~ m2 = tan (~ + ~r/2) then 2 2 m, + m2 = (Crx - ffy)/(p~xffy) tan c~ + tan (c~+~r/2) = Or] - cr2,)/(p%cry)
307
(40) (41) (42) (43)
simplifying tan 2e~ = -20%Cry/(Cr2x cry).
(44)
This simply shows that the slopes of the major and minor axes (comparing equations (21) and (44)) are determined in the reverse order. This also shows that normal least squares method is a better method and is related to the bivariate Gaussian distribution. The method of normal least squares gives a better understanding of the problem. In this method (fitting a general second degree equation for the ellipse) it is interesting to note that the step by step methods used first to determine a, b, c~ by the normal least squares method and the other method used for determining r~ and r2 are automatically combined to give the following identities. A = 1/~z~ B = 1/cry, f = -2p/(crx% ) u = (-2a&rZ~)+ (2pbArx%) v = (2pa&rxo-,.) - (2bhr2)
(45)
K = (a2/cr2
(50)
(46) (47) (48) (49)
2pabhrx% + b2/cr2y - 1)/(i-p2). DETERMINATION
OF ELLIPTICITY
To determine ellipticity eA or eB tWO concentric curves to ellipse enclosing the elliptical profile are constructed as explained below. For all practical purposes the ellipticity eA defined as the scaled difference along the semi-major axis of two concentric ellipses is better since such a construction is easy and also this definition gives least deviation of the profile of the ellipse compared to eB. The maximum and minimum normal deviations of each point from the best fit mean ellipse are calculated and their difference or absolute sum is ellipticity. To find the length d of the normal (perpendicular) from a point Q(xk, Yk) to the general point P(r~ cos 0, r2 sin 0) on the best fit ellipse the following method is found convenient. From Fig. 6, d = +- X/{(xk - rl
cos 0) 2 +
(Yk -- r2
sin 0) 2}
(51)
+ sign if (Xk, Yk) is outside the ellipse, - sign if (xk, Yk) is within the ellipse. The length of the normal is minimum of d and the position 0 is obtained by differentiating (51). This will give a fourth degree equation in sin 0. (~ _ ~)2 sin40 + 2 r~'k (r]l - ~ ) sin30 + [~ x~ - (r~ - ~)2 + ~ y]] sin20 - [2r~k ( d -- ~ ) l sin 0 -- ~ y~, = O.
(52)
With the four roots Ok determined from equation (52), they are substituted in equation (51) and least value of d is taken as the normal distance dk. From the normal deviations calculated at all points their maximum and minimum are obtained to draw the concentric ellipses or curves. The major and minor dia. of the
308
T . S . R . MURTHY
l x
FIG. 6. Orientation of principal axes of an ellipse.
6 I-
\,
o
" 30965
b
• 3.0923
r I = 3.013
r2 = 2.001 p = 0.502
CA= 0.2784
I
I
2
4
I 6
X
FIG. 7. Construction of two concentric ellipses to enclose the elliptical surface.
ellipse are modified by the s a m e a m o u n t as the deviations to construct the inner and o u t e r curves and ellipticity calculated. Figure 7 s h o w s the construction o f concentric ellipses e n c l o s i n g the elliptical profile for a specific case. Plus m a r k s in Fig. 7 indicate the m e a s u r e m e n t points o n the elliptical profile.
Methods for Evaluation of Elliptical Profiles
309
CONCLUSIONS
The analysis of the best fit ellipse presented in this paper gives an unambiguous values for ellipticity of elliptical profiles. The ellipticity eA defined on the basis of the construction of two concentric ellipses is convenient and easy to construct. However, the ellipticity e8 is large compared to eA. Three methods for the evaluation of elliptical profiles using: (1) normal least square solution; (2) bivariate Gaussian distribution; and (3) general second degree equation are described. Any of these methods can be used depending upon the users convenience. Using the recent developments in roundness measuring instruments with digital output the procedure described could be adopted with a built in program to provide circularity and or ellipticity directly. Thus these algorithms are directed towards the ultimate objective of developing efficient algorithms to enable development of software for computer aided metrology and manufacture. Acknowledgements--This work was carried out in the Department of Research and Testing, Central Machine Tool Institute, Bangalore. The author is thankful to the Director, CMTI, for permission to publish this work.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
R. E. REASON, Report on measurement of roundness, Rank Precision Industries Ltd. (1966). T. TSUKAOAet al., Proc. 19th IMTDR, pp. 529-535 (1978). M. GOTOand K. hZUKA,J. JSPE 41,477 (1982). T. S. R. MORTHV,Int. J. Mach. Tool Des. Res. 22,283-292 (1982). T. S. R. MORTnVet al., Int. Conf. on Metrology and Properties of Engineering Surfaces and Wear 57, 167-184. T. S. R. MtmTttV and S. Z. ABDIN, Int. J. Mach. Tool Des. Res. 20, 123-136 (1980). S. BRAI~D'r,Statistical and Computational Methods in Data Analysis, North Holland, Oxford (1978). T. S. R. MtmTHV. Proc. lOth AIMTDR, CMERI, Durgapur, India (1982). C. RAVICHANDRAand G. JAYARAMAN,Grinding the profile of three dimensional master cam for skrit turning of pistons, SME Technical Paper MR80-212 (1980). T. G. K. Mua'rHv, M. H. CVRtAC, G. SUDHEENDRAand M. MALAKONDAIAH.Optical Eng. 22, 602-606 (1983).
r2/F I
Ellipse as a locus of a point M (a) Such that the sum of the distances from two fixed points (foci) is constant i.e. F M + M F ' = 2r~ (b) Such that the ratio of the distances from a point (focus) and from a line (dircctrix) is constant and less th:m 1 i.e. F M / M K - ~ = < 1
COS I11 =
Ellipse as a section o f a (a) cylinder or (b) cone If ~ is the angle made between the horizontal and inclined planes then the coefficient of compression is given by
Ellipse as a compressed circle From any point N on the circle draw NP perpendicular to O X . The corresponding point M on the ellipse is such that PM/PN = r2/rl
Description
(a)
M
(a)
TABLE A I .
l, o
K
Y
Figure
M
(b)
(b)
Directrix
....
E L L I P S E A N D ITS DEF'INIFIONS
APPENDIX A
Used for construction of ellipses
T h e ellipse as a section o f a cylinder is m a d e use of, to evaluate ellipses using r o u n d n e s s m e a s u r ing i n s t r u m e n t s with long cylindrical stylus [8]
r2/r z = coefficient of compression of ellipse l - r z / r I = compression of ellipse rl/r 2 = axial ratio
Remarks
7o
Methods for Evaluation of Elliptical Profiles
311
TABLE A2. ELLIPSE AND ITS IMPORTANTPROPERTIES Description
1. The tangent drawn at a point (x~, y~) of the ellipse intersects the x-axis at the same point as the tangent of the circle x 2 + y2 = drawn at the points (xl, + X / ( ~ -
Figure
~
e
~
Tangentto (xf'y~)circle angent"to tipse
x"
x~)) Used to draw tangents and normals to ellipse
//NormaL
2. The tangent and the normal at a point of an ellipse, bisect the angle between the straight lines connecting that point to the loci (focal property) Used in a Spectrophotometer [101
K
3. The product of the distances between the loci and any tangent of an ellipse is constant and equals to ~ . The perpendicular drawn from either focus to a tangent meet the latter on the circle drawn with the major axis as dia.
Tangent Ira ipse
~
4. The two tangents drawn from any point N on the circle x 2 + y2 = + ~ and the ellipse x2/~ + y2/~ = 1 are at 90 ° Using to lap elliptical cams used in spectrophotometer [10]
Tangent 2 to elt~pse
312
T.S.R.
Muarnv
A P P E N D I X B: I N V A R I A N T S OF A G E N E R A L S E C O N D D E G R E E E Q U A T I O N When the second degree equation in x and y is written in the following form with % = a/i a t t x 2 + 2 a t 2 x y + a 2 ~ 2 + 2a~..dc + 2a23Y + a33 = 0
(BI)
or ( a l t x + az2y + al3) X + (a21 x + a , , 2 y + az3) Y + (a31x + a32..v + a33) = 0
(B2)
the three quantities all at2 ; A =la21 la,, a,2 a22 (B3) I a2l a22 la31 d32 a33 are called the invariaats o f the second degree equation. Since a 0 are known in terms of A, B, f,, u and v, the quantities 1, 8, A are calculated. After proper rotation and translation of the axes x and y, the equation (B1) transforms to I = all + a22 ; ~ =
dll X2 4" a22Y2 -t- d33 = 0 with unknows a~,, a22, 1~33 then the invariants o f the equation (B4) are 1 = ajl + a22 ; ~ --- aj~ a22 ; A = alla22a33 , By solving (B3) and (B5) the unknown art, d22, a3~ are obtained as ,i~3 = a / ~ ; e,,.2., = ( X ± V ( ~
(B4)
(B5)
- 48))/2.
The centre (a,b) of the ellipse is obtained by solving the equations of major and minor axes. They are given by equating to zero the bracketed terms namely the coefficients of x and y in equation (B2) namely a~x + a~ + al3 = 0 a2~x + a,,2y + a-,3 = O.
Vol. 25, No. 4. pp. 299--312, 1985.
0020-7357/8553.00+.00 Pergamon Press Ltd.
METHODS FOR EVALUATION OF ELLIPTICAL PROFILES T. S. R. MURTHY* (Received 6 September 1982; in final form 5 December 1984)
Abstract--There are many interesting properties of elliptical profiles used in precision instruments which simplify the mechanisms to obtain a desired function. One such application is a spectrophotometer developed at CMTI. The other applications are elliptical gear blanks, pistons of I.C. engines etc., whose cross sections are elliptical. Elliptical sections having very little difference between major and minor dia. can be measured with the roundness measuring instruments to obtain the difference between the major and minor dia. However, fitting of best fit ellipse and hence the definition of eilipticity are not available. In this paper 3 different methods are proposed for the evaluation of the elliptical profiles, based on normal least squares fit, bivariate Gaussian distribution and general second degree equation. In the 3rd method the deviations from the property of the ellipse are minimised.
NOMENCLATURE A B D E,, E~
N ai a
b C
dr dk eA,eB
f
h ml,m2
P i ,P2 r I ,r2
b/ v
x',y' xi,Yi xp,yp
0 P ~x ~v ~xy
+(x,y)
constant (coefficient of x 2) in the general second degree equation in x and y constant (coefficient of y2) in the general second degree equation in x and y constant in the general second degree equation in x and y sum of the squares of the normal deviation sum of the squares of the deviation from the property of the ellipse number of approximate equi-angular spaced data points constants in the general second degree equation in x and y X-coordinate of the centre of the best fit ellipse Y-coordinate of the centre of the best fit ellipse constant in the equation of a straight line y = mx + c distance from a point (xk,Yk) to a point on the ellipse normal distance from measurement point k on to the best fit ellipse ellipticity values constant (coefficient of xy) in the general second degree equation in x and y difference in the semi-major and semi-minor dia. of two concentric ellipses slope of the major axis and slope of the minor axis of the best fit ellipse semi-major and semi-minor dia. of an ellipse (representing constant probability) semi-major and semi-minor dia. of the best fit ellipse constant (coefficient of x) in the general second degree equation in x and y constant (coefficient of y) in the general second degree equation in x and y local coordinate axes passing through the centre and along the major axis and minor axis of the best fit ellipse x and y coordinates of the ith measured point on the ellipse x and y coordinates of a point on the curve parallel to the ellipse orientation of the major axis, tan a = m angle made by the normal (to the ellipse at a point on it) with the x-axis polar coordinate correlation coefficient standard deviation of x standard deviation of y co-variance (x, y) joint probability distribution INTRODUCTION
MEASUREMENTand evaluation of circular profiles of cylindrical and spherical components in terms of circularity [1], cylindricity [2-4] and sphericity [5] has been well established by the use of roundness measuring instruments. Elliptical sections having very little difference between major and minor dia. can be measured with roundness measuring instruments to obtain the difference between their dia. However methods for fitting the best fit ellipse and definition of ellipticity (form deviation of elliptical profile) are not available. * Central Machine Tool Institute. Bangalore 560 022, India. 299
300
T . S . R . MURTHY
There are many interesting and useful applications for elliptical profiles in engineering practice used in instruments like a spectrophotometer developed at CMTI for obtaining various incidence angles between 2° and 60°. The other applications are pistons of IC engines (a special manufacturing technology developed at CMTI [9]) and elliptical gear blanks. Since no method for evaluation of elliptical profiles are available an attempt has been made in this paper to define its form deviation namely ellipticity and devise methods for its evaluation. Three methods for the construction of the best fit ellipse are proposed. The first method is called a Normal Least Squares Method which simultaneously fits the major and minor axis to the data corresponding to ellipse. In the second method the measured x and y coordinates of the elliptical profile are assumed to follow bivariate Gaussian distribution and hence the parameters of this distribution are related to the constants in the equation of the ellipse namely the centre of the ellipse and the orientation of the major axis. The third method fits a general second degree equation in x and y to the data by minimizing the sum of the squares of the deviation from the property of the ellipse. These methods are directed towards the ultimate objective of developing efficient algorithms for computer aided metrology and manufacture. ELLIPSE AND ITS PROPERTIES
Ellipse can be defined and understood in different ways. Appendix A shows some of the definitions, important properties of the ellipse and their engineering application. The standard canonical form of the ellipse is given by x 2/ ~ ") + y 2 / ~ =
1.
DEFINITIONS
The term ellipticity is generally used to mean the deviation from the form of a circle or sphere. The coefficient of compression of ellipse (Appendix A) is rCr2, its reciprocal is called axial ratio and ellipticity in mathematics is defined as 1-r2/r~. But for engineering metrology application the term ellipticity has been defined similar to circularity to represent the deviation from the form of an ellipse. For defining ellipticity construction of inner and outer parallel curves to ellipse is helpful. In the polar form the equation of the ellipse shown in Fig. 1 is given by x = rt cos 0 y = r2 sin O.
(1)
If 13is the angle made by the normal at a point on the ellipse the locus of the point with constant distance h from the ellipse is given by xp = rl cos O + h cos~3
yp = r2 sin 0 + h sin 13
(2)
where 13= tan-~ (rl tan O/r2) is the angle made by the normal to the ellipse at any point on it. (Tan 13 = - d x / d y is obtained by differentiating equation (1).) Equation (2) is not an ellipse with semi-major and minor dia. increased by h. If the semi-major and minor dia. are increased by h the corresponding equation of the ellipse would be x = (rl + h) cos0 y = (r2 + h) sin 0.
(3)
Figure 2 shows the plot of equations (1), (2) and (3) with h positive for outer curve and h negative for inner curve to the ellipse given by equation (1). From Fig. 2 the
M e t h o d s for E v a l u a t i o n of E l l i p t i c a l Profiles Locus o f a
t Y
-- -- -- -- -- /
301
paraLLeL curve x p, y . )
h
\
\
o rl cos 8
;I
I-
FI6. 1. C e n t r a l ellipse and the locus of a p o i n t with c o n s t a n t n o r m a l d i s t a n c e , h.
~Parattet \equation
curve "to eLLipse ( 2 ) with negative
__
tY ~
,
L_ equation ( 2 ) with 'h' positive
ti
x2
EL pse i7~~_h~2 ,-i--,
+
y2
,._;-'SZT2 ,,~,--,
....
tr~n~
¢t/n~=
o x~
o Y" -
,,..++ -'~-~ + -~-z~ - I x
~r tn+
F1G. 2. E l l i p s e and d i f f e r e n t p a r a l l e l c u r v e s to ellipse.
following points should be noted: (a) Equation (2), namely parallel curve to the ellipse, and equation (3), namely an ellipse with equal increase in semi-major and minor dia., are different. Both of them meet at 4 points i.e., at the extremities of major and minor dia. when outer curve is considered. The curve given by equation (2) is continuous when h is positive. (b) With h negative a discontinuous inner curve is obtained with fish tail shape at the extremities of the major axis. (c) The area enclosed between the inner and outer curves given by equation (2) is larger compared to the area enclosed between the inner and outer ellipses given by equation (3) with same h. Because of the above difference between equations (2) and (3) two definitions for ellipticity are brought out: Ellipticity eA: is defined as the scaled difference along the semi-major axis of two concentric ellipses constructed around one of the specified best fit ellipses which are just sufficient to contain the total elliptical profile. Ellipticity eB: is defined as the scaled difference along the semi-major axis of two concentric curves constructed parallel to one of the best fit ellipses which are just
302
T.S.R.
MURTHY
sufficient to contain the total elliptial profile (Fig. 2). The centre of the best fit ellipse and the orientation of the major and minor axes are constructed so as to satisfy any one of the following: (a) Minimum Zone (separation) Ellipse (MZE) is the mean ellipse which gives ellipticity eA or ea minimum. (b) Least Squares Ellipse (LSE) is the mean ellipse satisfying the least squares condition for the property of the ellipse. (c) Maximum Inscribed Ellipse (MIE) is the largest ellipse that can be inscribed within the elliptical profile. (d) Minimum Circumscribed Ellipse (MCE) is the smallest ellipse that contains the elliptical profile. The minimum zone ellipse will be the preferred best fit ellipse because it gives least error. Since MZE has to be found by search techniques, [6] requires a starting solution. Hence methods for finding the centre and orientation of the best fit ellipse have been dealt with in this paper. METHODS OF EVALUATION
When the measurements are taken at specified points on the ellipse, then the ellipse in engineering terms is represented by the measured points itself in rectangular or polar coordinates. As the coordinate system can be easily interchanged it is assumed that the coordinates of measured points on the ellipse are available for analysis. (a)
M e t h o d o f n o r m a l least squares
This method determines the centre of the ellipse and the orientation of major and minor axes. Considering measured points xi, Yi to be available (Fig. 3) on the elliptical profile it is possible to fit for the measured data a general least squares straight line of the form y = mx + c to represent the major axis by minimizing the sum of the squares of the deviations (~,PM2 = Z ( y i - y ) 2 ) . Similarly it is possible to fit a general least squares line of the form x = m ' y + c' to represent the minor axis. If these two lines are mutually perpendicular then they can be taken to represent the major and minor axes. This means ram' = - 1.
~ J
•ly:m2 JMinoraxis x +%
\
Majoraxis
\ Y 23 ~
0
FIG.3.
I
I
I
y~)
N
I
I
2
3
/1
~rla'b)
I
4 X
~
~ll~ 5
2: 190+ a~). I I 6
7
Normal least squares method for fitting the axis of the ellipse.
Methods for Evaluation of Elliptical Profiles
303
But in the general least squares methods the product m m ' i s not exactly - 1 , but m m ' = r~ = correlation coefficient which lies between 0 and + 1. A method which automatically satisfies the condition that the product of the slopes of the straight lines representing the major axis and minor axis is - 1 (ram' = - 1 ) is called normal least squares method by the author and has been first used in finding the minimum zone evaluation of surfaces [6] and extended now to determine the orientation of the axes of the elliptical profiles. In the normal least squares method the sum of the squares of the perpendicular distances (Fig. 3) from a point P (xi, Yi) on to the straight line of the form y = m x + c are minimized. (4)
En = £ P N z = £[(Yi-Y) cos ot] 2.
It is shown below how two mutually perpendicular straight lines are fitted by normal least squares method from the above equation. Substituting y = taxi + c and cos 2 t x = l / ( l + m 2) in equation (4) gives
(s)
E,, = Y ~ ( y i - m x i - c ) 2 / ( l +m2).
Differentiating partially equation (5) with respect to m and c and equating to zero the equations to solve for m and c are
{[N~x 2 -- (Y, Xi) 21 -- [NZy 2 - (£yi)2] } m2 + m
[N'ZxiYi
~ j
and c = Y. ( y i - m x i ) / N .
- 1 = 0
(6) (7)
Equation (6) is quadratic in m (and has two solutions ml, m2) and the constant term is - 1 (product of the roots m m2 = - 1 ) . Hence equation (6) gives the slopes of the two mutually perpendicular straight lines fitted by normal least squares method. The intersection of these two mutually perpendicular straight lines gives the centre (a,b) of the ellipse as derived below. If these two straight lines are given by y = m ~ x + c l and y = m 2 x + c 2 ; substituting a for x and b for y and solving for a and b a = (c 1 - c 2 ) / ( m
l+m2)
substituting cl, c2, ml, m2 from equations (6) and (7) a = Exi/N
and
b = Y.yi/N.
(8) (9)
Since a and b are average locations along x and y axes the solution for a, b, mr, m2 are suitable for the construction of the best fit ellipse. (b)
M e t h o d o f joint probability distribution
This method is applied here mainly to determine the centre (a,b) of the ellipse and the orientation of the axes. In this method the property of the joint normal distribution of two variables x and y (bivariate Gaussian distribution) which shows that the curves of constant probability are ellipses is made use of to determine the centre and orientation of the ellipse.
~i~oR 2 5 : 4 - B
304
T.S.R.
MUllTH¥
The joint normal distribution (Fig. 4) of the two variables x and y is given by [7]
O0(x,y) =
2~rex~y X / ( l - p 2) exp
x-a
.l:xa:ox,
[
y-b
2(1_p2 )
{ y-b
~2l]
) ( W. ) + ," "Ho--V,
where p = cov (x,y)/(cr,~ry) = Cr2x/(~rxCry).
(10)
(11)
In equation (10) the curves of constant probability are determined by equating the exponent to a constant K. i.e.
x--a
, ox
(y-b
)
(y-b] 2
+ ,o,,--
--
This is the equation of an ellipse with centre (a,b) as shown in Fig. 5 with parameters ~.~, ~v and 9. The principal axis of the ellipse makes an angle a with respect to the x-axis as shown in Figs 3 and 5. The angle e~ and the semi dia. Pt and P2 are derived from the well known properties of conic sections as tan 2e~ =
p2 =
2 p ~ x O r y / ( C r x2 --ffy) 2
(13)
2 2 2 2 ot--29CrxCrysin a cos a + Crx 2 sin2tx) crxcrrK(19 2 )/(CryCOS
p2 = cr~o~yK(a_p2)/(crZsinZa+2pCrx%,
sin a cos a + ~ cosZoO.
(14) (15)
Since we are interested in the centre (a,b) of the ellipse given by equation (12) and orientation a, their maximum likelihood estimates are given by
a = Exi/N = £ b = Eyi/N = 37 O"x2
y~(xi_a)2/N
2 O'y
= Z(yi_b)2/N p = X(xi-a) ( y i - b ) / ( N c r x ~ ) a n d 2 "~ tan 2a = 2pCrxCry/(Crx-~7.).
(16) (17) (18) (19) (20) (21)
Determination of major and minor d&. After locating the centre and orientation of the ellipse by the above methods semi-major and minor dia. have to be determined to construct the best fit ellipse. This requires a least squares criterion which is slightly different from the general least squares method and is described below. For convenience and simplicity the coordinates of the data points (xi, Yi) a r e measured with respect to a new coordinate system (x',y') with the origin at (a,b) and x-axis along the major axis so that the equation of the ellipse now is of the form x,2/~ + y,Z/~ = 1.
(22)
If a point (x[, y[) lies on the ellipse then equation (22) will be exactly satisfied, i.e., left hand side term will be exactly unity. If it is not on the ellipse the left hand side term will not be exactly unity. The sum of the squares of the deviation from unity has been minimized for the solution. These deviations are named as the deviations from the property of the ellipse and are given by
Es = Z(x:Z/r 2 + y:2/~ _ 1)2
(23)
Methods for Evaluation of Elliptical Profiles
305
~(x,y)
FIG. 4. Probability density of a bi-variate Gaussian distribution.
For K=I
Moior a x i s /
3.6 3.5-2.8-2.4-Y 2-1,6-1.2-0.8 -0.4 --
I
0.4
08
I
I
12
I
15
2
I
2.4
I
2.8
I
32
X
FIG. 5. Co-variance ellipse.
= X(Ax; 2 + By 2 - 1) 2 where A = 1/~ and B = 1/~.
(24)
Differentiating equation (23) partially with respect to A and B and equating to zero the equations to determine A and B and hence rl, r2 in matrix form are given by
X//4
Xx;2y;2
A Xy;2 ] .
(25)
(c) Method of best fit general second degree equation in x and y Methods (a) and (b) are a step by step approach to determine first a, b, a and then rl and re. In this method a general second degree equation in x and y of the form given below is assumed.
306
T.S.R.
A'x 2 + B'y 2 +
MURTHY
f x y + u'x + v'y - D = 0.
(26)
If D :k 0, equation (26) is rewritten by dividing by D as Ax 2
+ By 2 + fxy + ux + vy - 1 = 0
(27)
where A = A ' / D etc. T h e sum of the squares of the deviation from the p r o p e r t y of the ellipse (conic section) is given by E, = (Ax 2 + By2i
+
fxiYi + uxi +
vyi --
1) 2.
(28)
Differentiating partially with respect to A, B, f, u, v and equating to zero the equations in the matrix form to d e t e r m i n e A, B, f, u, v, are given by equation (29).
x~4
Xx~
Xx~, Xx~
x~,
2x~
Exi2y2
Ey~
Exiy 3
Ey~
Yy~
x@,
Xx~v~ x@ 2 Xx~,
Xx,y?
Ex,~¢i
Ex 3
Ex,y2~ Ex~,
Ex~
Ex,yi
Yxi
Xxi2Yi
Xy 3
Exzyi
E,y:i
Zxiy 2
Exiy 2
v
Eyi
(29)
I
J
F r o m A, B, f, u, v the diameters of the ellipse and its centre etc. are obtained as (30) (31) (32)
a = ( v f - 2 B u ) / ( 4 A B - f 2) b = ( u f - 2 A v ) / ( 4 A B - f 2) tan 2~ = f / ( A - B ) - - u v f + ( A v 2 + B u 2) +
4(AB -f2/4)
~' ~ = 23~-B~f~ i ~ + ~ - ' ~ ~
"
(33)
T h e s e quantities are e v a l u a t e d with the aid of invariants of the second o r d e r curve as shown in A p p e n d i x B. COMPARISON
It is interesting to note that there exists some relation among the three methods. In the m e t h o d of n o r m a l least squares using the relations 2
(~ X
Similarly
~
E(Xi
X)2/N
or [NEx 2 - (Exi) 2] = U2cr2. • [N'Zy 2 - (Eyi) 2] = U2cr~, (N~,xiy i - ~,xi~,,yi) = N 20.xy 2
and substituting t h e m in e q u a t i o n (6) m 2 + m [(0.2 - 0.~)hrxy] 2 2 - 1 = 0
(34)
(35) (36) (37)
(38)
using 2
D = (Txy/ITxffy, w e get
m 2 + m [(or2 - trz)/(pcr.,cr:.)] - 1 = 0.
(39)
M e t h o d s for E v a l u a t i o n of E l l i p t i c a l Profiles
In the above equation since mira 2 = - 1 if m I = tan o~ m2 = tan (~ + ~r/2) then 2 2 m, + m2 = (Crx - ffy)/(p~xffy) tan c~ + tan (c~+~r/2) = Or] - cr2,)/(p%cry)
307
(40) (41) (42) (43)
simplifying tan 2e~ = -20%Cry/(Cr2x cry).
(44)
This simply shows that the slopes of the major and minor axes (comparing equations (21) and (44)) are determined in the reverse order. This also shows that normal least squares method is a better method and is related to the bivariate Gaussian distribution. The method of normal least squares gives a better understanding of the problem. In this method (fitting a general second degree equation for the ellipse) it is interesting to note that the step by step methods used first to determine a, b, c~ by the normal least squares method and the other method used for determining r~ and r2 are automatically combined to give the following identities. A = 1/~z~ B = 1/cry, f = -2p/(crx% ) u = (-2a&rZ~)+ (2pbArx%) v = (2pa&rxo-,.) - (2bhr2)
(45)
K = (a2/cr2
(50)
(46) (47) (48) (49)
2pabhrx% + b2/cr2y - 1)/(i-p2). DETERMINATION
OF ELLIPTICITY
To determine ellipticity eA or eB tWO concentric curves to ellipse enclosing the elliptical profile are constructed as explained below. For all practical purposes the ellipticity eA defined as the scaled difference along the semi-major axis of two concentric ellipses is better since such a construction is easy and also this definition gives least deviation of the profile of the ellipse compared to eB. The maximum and minimum normal deviations of each point from the best fit mean ellipse are calculated and their difference or absolute sum is ellipticity. To find the length d of the normal (perpendicular) from a point Q(xk, Yk) to the general point P(r~ cos 0, r2 sin 0) on the best fit ellipse the following method is found convenient. From Fig. 6, d = +- X/{(xk - rl
cos 0) 2 +
(Yk -- r2
sin 0) 2}
(51)
+ sign if (Xk, Yk) is outside the ellipse, - sign if (xk, Yk) is within the ellipse. The length of the normal is minimum of d and the position 0 is obtained by differentiating (51). This will give a fourth degree equation in sin 0. (~ _ ~)2 sin40 + 2 r~'k (r]l - ~ ) sin30 + [~ x~ - (r~ - ~)2 + ~ y]] sin20 - [2r~k ( d -- ~ ) l sin 0 -- ~ y~, = O.
(52)
With the four roots Ok determined from equation (52), they are substituted in equation (51) and least value of d is taken as the normal distance dk. From the normal deviations calculated at all points their maximum and minimum are obtained to draw the concentric ellipses or curves. The major and minor dia. of the
308
T . S . R . MURTHY
l x
FIG. 6. Orientation of principal axes of an ellipse.
6 I-
\,
o
" 30965
b
• 3.0923
r I = 3.013
r2 = 2.001 p = 0.502
CA= 0.2784
I
I
2
4
I 6
X
FIG. 7. Construction of two concentric ellipses to enclose the elliptical surface.
ellipse are modified by the s a m e a m o u n t as the deviations to construct the inner and o u t e r curves and ellipticity calculated. Figure 7 s h o w s the construction o f concentric ellipses e n c l o s i n g the elliptical profile for a specific case. Plus m a r k s in Fig. 7 indicate the m e a s u r e m e n t points o n the elliptical profile.
Methods for Evaluation of Elliptical Profiles
309
CONCLUSIONS
The analysis of the best fit ellipse presented in this paper gives an unambiguous values for ellipticity of elliptical profiles. The ellipticity eA defined on the basis of the construction of two concentric ellipses is convenient and easy to construct. However, the ellipticity e8 is large compared to eA. Three methods for the evaluation of elliptical profiles using: (1) normal least square solution; (2) bivariate Gaussian distribution; and (3) general second degree equation are described. Any of these methods can be used depending upon the users convenience. Using the recent developments in roundness measuring instruments with digital output the procedure described could be adopted with a built in program to provide circularity and or ellipticity directly. Thus these algorithms are directed towards the ultimate objective of developing efficient algorithms to enable development of software for computer aided metrology and manufacture. Acknowledgements--This work was carried out in the Department of Research and Testing, Central Machine Tool Institute, Bangalore. The author is thankful to the Director, CMTI, for permission to publish this work.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
R. E. REASON, Report on measurement of roundness, Rank Precision Industries Ltd. (1966). T. TSUKAOAet al., Proc. 19th IMTDR, pp. 529-535 (1978). M. GOTOand K. hZUKA,J. JSPE 41,477 (1982). T. S. R. MORTHV,Int. J. Mach. Tool Des. Res. 22,283-292 (1982). T. S. R. MORTnVet al., Int. Conf. on Metrology and Properties of Engineering Surfaces and Wear 57, 167-184. T. S. R. MtmTttV and S. Z. ABDIN, Int. J. Mach. Tool Des. Res. 20, 123-136 (1980). S. BRAI~D'r,Statistical and Computational Methods in Data Analysis, North Holland, Oxford (1978). T. S. R. MtmTHV. Proc. lOth AIMTDR, CMERI, Durgapur, India (1982). C. RAVICHANDRAand G. JAYARAMAN,Grinding the profile of three dimensional master cam for skrit turning of pistons, SME Technical Paper MR80-212 (1980). T. G. K. Mua'rHv, M. H. CVRtAC, G. SUDHEENDRAand M. MALAKONDAIAH.Optical Eng. 22, 602-606 (1983).
r2/F I
Ellipse as a locus of a point M (a) Such that the sum of the distances from two fixed points (foci) is constant i.e. F M + M F ' = 2r~ (b) Such that the ratio of the distances from a point (focus) and from a line (dircctrix) is constant and less th:m 1 i.e. F M / M K - ~ = < 1
COS I11 =
Ellipse as a section o f a (a) cylinder or (b) cone If ~ is the angle made between the horizontal and inclined planes then the coefficient of compression is given by
Ellipse as a compressed circle From any point N on the circle draw NP perpendicular to O X . The corresponding point M on the ellipse is such that PM/PN = r2/rl
Description
(a)
M
(a)
TABLE A I .
l, o
K
Y
Figure
M
(b)
(b)
Directrix
....
E L L I P S E A N D ITS DEF'INIFIONS
APPENDIX A
Used for construction of ellipses
T h e ellipse as a section o f a cylinder is m a d e use of, to evaluate ellipses using r o u n d n e s s m e a s u r ing i n s t r u m e n t s with long cylindrical stylus [8]
r2/r z = coefficient of compression of ellipse l - r z / r I = compression of ellipse rl/r 2 = axial ratio
Remarks
7o
Methods for Evaluation of Elliptical Profiles
311
TABLE A2. ELLIPSE AND ITS IMPORTANTPROPERTIES Description
1. The tangent drawn at a point (x~, y~) of the ellipse intersects the x-axis at the same point as the tangent of the circle x 2 + y2 = drawn at the points (xl, + X / ( ~ -
Figure
~
e
~
Tangentto (xf'y~)circle angent"to tipse
x"
x~)) Used to draw tangents and normals to ellipse
//NormaL
2. The tangent and the normal at a point of an ellipse, bisect the angle between the straight lines connecting that point to the loci (focal property) Used in a Spectrophotometer [101
K
3. The product of the distances between the loci and any tangent of an ellipse is constant and equals to ~ . The perpendicular drawn from either focus to a tangent meet the latter on the circle drawn with the major axis as dia.
Tangent Ira ipse
~
4. The two tangents drawn from any point N on the circle x 2 + y2 = + ~ and the ellipse x2/~ + y2/~ = 1 are at 90 ° Using to lap elliptical cams used in spectrophotometer [10]
Tangent 2 to elt~pse
312
T.S.R.
Muarnv
A P P E N D I X B: I N V A R I A N T S OF A G E N E R A L S E C O N D D E G R E E E Q U A T I O N When the second degree equation in x and y is written in the following form with % = a/i a t t x 2 + 2 a t 2 x y + a 2 ~ 2 + 2a~..dc + 2a23Y + a33 = 0
(BI)
or ( a l t x + az2y + al3) X + (a21 x + a , , 2 y + az3) Y + (a31x + a32..v + a33) = 0
(B2)
the three quantities all at2 ; A =la21 la,, a,2 a22 (B3) I a2l a22 la31 d32 a33 are called the invariaats o f the second degree equation. Since a 0 are known in terms of A, B, f,, u and v, the quantities 1, 8, A are calculated. After proper rotation and translation of the axes x and y, the equation (B1) transforms to I = all + a22 ; ~ =
dll X2 4" a22Y2 -t- d33 = 0 with unknows a~,, a22, 1~33 then the invariants o f the equation (B4) are 1 = ajl + a22 ; ~ --- aj~ a22 ; A = alla22a33 , By solving (B3) and (B5) the unknown art, d22, a3~ are obtained as ,i~3 = a / ~ ; e,,.2., = ( X ± V ( ~
(B4)
(B5)
- 48))/2.
The centre (a,b) of the ellipse is obtained by solving the equations of major and minor axes. They are given by equating to zero the bracketed terms namely the coefficients of x and y in equation (B2) namely a~x + a~ + al3 = 0 a2~x + a,,2y + a-,3 = O.