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Hermite interpolation with a pair of spirals
Computer Aided Geometric North-Holland
COMAID
Design
10 (1993) 491-507
491
322
Hermite interpolation of spirals *
with a pair
D.S. Meek Department of Computer Science, Unk,ersity of Manitoba, Winnipeg, Canada
R.S.D. Thomas St. John T Collrge and Department of Applied Mathematics, Unicersity of Manitoba. Winnipeg, Canada Received July 1991 Revised December 1992
Abstract Meek. D.S. and R.S.D. Thomas, Design 10 (1993) 491-507.
Hermite
interpolation
with a pair of spirals,
The Hermite interpolation problem in the plane considered tangent vectors and signed curvatures at the two points with The rotation of the tangent vector of the interpolating curve less than r. The necessary and sufficient conditions for the Keywords. Hermite
interpolation;
Computer
Aided
Geometric
here is to join two points and to match given unit various G2 curves consisting of a pair of spirals. from one point to the other is restricted to being existence of each of the various curves are given.
spirals.
1. Introduction
The unit tangent vector is a unit vector that is parallel to the tangent line and that points in the direction of increasing arc length. Curvature is a signed quantity which is positive if the centre of curvature is on the left when moving in the direction of increasing arc length. A spiral is a curve with monotone curvature of constant sign. A G* curve is a continuous curve with continuous unit tangent vector and continuous curvature. The Hermite interpolation problem is to find a curve which passes through given points and also matches higher-derivative data such as tangents and curvatures at those points. A biarc, a pair of circular arcs joined with matching tangent, can be used as a Hermite element that passes through two given points and matches unit tangents at the two points [Nutbourne & Martin ‘88, Su &Liu ‘891. A curve made of up to four clothoids can be used as a G* Hermite element that matches unit tangent vectors and curvatures at its endpoints [Pal & Nutbourne ‘771. A pair of conic segments can also be used as a G2 Hermite element that matches unit tangents and curvatures at its endpoints [Pottmann ‘911. The planar Her-mite interpolation problem considered here is to join two given distinct points P and Q, to match given unit tangent vectors, t,, tQ, and signed curvatures K,, K,, at Correspondence to: D. Meek, Department of Computer Science, University of Manitoba, Winnipeg (Man.). Canada R3T 2N2. Email addresses of the authors, Dereck_ [email protected] and Robert [email protected]. * This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada. 0167~8396/93/506.00
D 1993 - Elsevier
Science
Publishers
B.V. All rights reserved
192
D.S. .Cleek. R.S.D. Thomas / Pair of spirals
Fig. 1. Data in standard
form with radii of curnature
indicated.
P and Q with various G’ curves consisting of a pair of spirals. The new results presented here generalize Guggenheimer’s results for interpolation with a single spiral [Guggenheimer ‘631.A standard form of the interpolation problem is obtained by placing P at the origin, aligning t, along the positive r-a.xis, and reflecting across the x-axis if necessary so that K, 2 0 (Fig. 1). Let the rotation of the tangent vector from t, to tQ be W. The angle W is restricted, 0 < W < T, because the results for a spiral in [Guggenheimer ‘631 break into two cases depending on whether the angle is less than or greater than r. The restriction on W is sensible as one would not normally want the curve to rotate too much between two interpolation points, and it means that the curves will not self-intersect or form loops. The necessary and sufficient conditions for the existence of a single spiral joining P to Q and matching t,, tQ, K,, K, have been studied in [Guggenheimer ‘631 and [Meek & Walton ‘921. The conditions are summarized in Section 2 of this paper. Of the possible G2 curves consisting of a pair of spirals, the following three GZ curves in standard form are the only ones that need be considered: a C-curve whose curvature decreases monotonically to a nonnegative minimum and then increases monotonically; a U-curve whose curvature increases monotonically to some given maximum and then decreases monotonically; and an S-curve whose curvature decreases monotonically and becomes negative. The above curves will be discussed in Sections 3-5. The reason for studying curves made of a pair of spirals is that such curves have considerable flexibiIi~, but at the same time are somewhat restrictive. For example, a C-curve or a U-curve has exactly one curvature extremum and no inflection points, while an S-curve has exactiy one inflection point. Spirals cannot self intersect or form cusps; commoniy used parametric cubic curves can [Wang ‘811. Spirals can be bounded by circular arcs since the spiral from any point is entirely inside or entirely outside the circle of curvature at that point [Guggenheimer ‘63, p.481 With each of the curves made from a pair of spirals, there exist choices of Q, W, K,, and K, such that there is no curve of that type joining P to Q. For example, it is impossibie to find a C-curve of positive curvature (and the rotation of the tangent vector restricted to less than r;) joining P to Q if Q is below the x-axis. Given W, K,, K,, a region of the plane will be described such that the curve exists if and only if Q is in the interior of that region. If it seems strange that an interpolation curve may not exist, recall that interpolation between two points matching unit tangent vectors and signed curvatures at the two points is not always possible even with parametric cubic curves [BCzier ‘72 (p. 1231, de Boor et al. ‘87, Goodman & Unsworth ‘881. For example, if the curvatures at the two end points P and Q of a cubic BCzier curve are zero, the Bezier curve is a straight line joining P to Q, and the tangents are determined. Thus, these tangents may not match given unit tangent vectors at P and Q.
D.S. kiwk. R.S.D. Thomas / Pair of spirals
493
2. Single spiral
The problem of using a single spiral to join two points. matching curvatures and unit tangent vectors at those points, is discussed in [Guggenheimer ‘631. The following is a summary of Guggenheimer’s results and those in [Meek Rr Walton ‘921. Assume that the spiral is not a straight line segment or circular arc, or that K, f K,. There are four cases: nondecreasing curvature starting with (a) zero and with (b) positive curvature, and nonincreasing curvature ending with (c) zero and with (d) positive curvature. Case (a). Consider spirals with nondecreasing curvature and starting with zero curvature, or 0 = K,
-40))
fxn
+.+2(W) (J
-n(O)),
(2.1)
where
and s > 0, x > 0. The vector n(e) forms an angle of 0 - 7r/2 with the positive x-axis. The expression (2.1) can be thought of as a point plus the convex sum of two vectors. Thus, the region described is a wedge-shaped region with apex at the point (l/KQ)(n(W) -n(O)) and boundaries obtained when s = 0 and x = 0 in (2.1) (Fig. 2). Case (b) Consider spirals with nondecreasing curvature and starting with positive curvature, 0
Fig. 2. Region
associated
with a spiral with 0 = K, < Kp.
D.S. Meek. R.S. D. Thomas / Pair of spirals
Fig. 3. Region
associated
with a spiral with 0 < K, < KQ.
Case (c) Consider spirals with nonincreasing curvature. If the spiral ends with zero curvature, 0 = K,
&w P
(
+w -40)) +xn -f?T
) +s(n(Jv)
-n(O)),
(2.3)
where s 2 0, x 2 0. The above expression can be thought of as a point plus the convex sum of two vectors. Thus, the region is a wedge-shaped region with apex at the point (l/K,)(n( W) n(O)) and boundaries obtained when x = 0 and s = 0 in (2.3) (Fig. 4).
Fig. 4. Region
associated
with a spiral with 0 < Kp < K,.
D.S. Meek, R.S. D. Thomas / Pair of spirals
Fig. 5. Region
associated
with a spiral with 0 < K, < K,.
Case (d) Consider spirals with nonincreasing curvature and ending with positive 0
curvature, and unit
(2.4) where 0 K, and K, < K, lie on opposite sides of the ray through P parallel to n(W) - n(O).
3. c-curve The C-curve considered here is a G’ curve made by joining a pair of spirals of nonnegative curvature so that the curvature of the C-curve decreases from a positive K, at P to a minimum value, K,, then increases to a positive K, at Q. Either end of the C-curve could be taken as the starting point; so, without loss of generality, assume K, a K,. Let M be a point of minimum curvature, and let the angle that the unit tangent vector at M makes with the x-axis be 8, 0 < 8 < W. The analysis depends on whether K, is zero or positive. The region for K, = 0 includes the region for K, > 0,and is its limit as K,${ is allowed to approach zero. The case K, = 0 is treated first because it is simpler. Case 1. First, consider the case when the minimum curvature is 0, 0 = spiral from P to M, by (2.31, has M(s, x, 0) in the interior of the region
~0 P
T+e -40)) +xn (r
) +.+(e)-n(o)),
K, < KQ G K,.The
~2=0,~20.
496
D.S. Meek. R.S.D. Thomas / Pair of spirals
Fig. 6. Wedge
associated
with a C-curve,
K,, = 0.
Now, considering the spiral from M to Q and using (2.1) (rotate the region (2.1) by 0 and translate its origin to MI, then Q can be in the interior of the region M(s, x, S) +R(e)
i
&w-e)
-n(O))
+yn ;
KQ
i
j
+t(n(w-e)
-n(o))
where t 2 0, y a 0, 0 G 13d W, and R(B) is the rotation matrix R(B)
=
-sin “). cos I3
(?,”
The expression for this region can be thought of as a point
(3.1) where S = l/K,
- l/K,, ,?r
plus the convex sum of three vectors \
(3.2) where s > 0, t a 0, x 2 0, y a 0, 0 Q 0 Q W. The first vector forms an angle of 8 with the x-axis, the second vector forms an angle of e/2 with the x-axis, and the third vector forms an angle of (0 + WI/2 with the x-axis. Since 0 Q 8 < W, the first vector is redundant as it is between the other two vectors; without loss of generality, set x = y = 0. Thus, for each 8, the region is the interior of a wedge shape with apex A(B) and bounded by lines that form angles f3/2 and (0 + W)/2 with the x-axis (Fig. 6). As A(8) moves around its circular path from F =A(O) to G =A(W), the arm of the wedge at angle 8/2 always passes through F, and the extended arm at angle (0 + W)/2 always passes through G. Thus, it is obvious that the wedge sweeps out the nonconvex region shown in Fig. 7. The boundaries of the region are obtained from (3.1) and (3.2) with x = y = 0: the ray at angle W when s = 0 and 8 = W, the line segment at angle W/2 when 0
D.S. Meek. R.S. D. Thomas
/ Pair o/spirals
Fig. 7. Region associated with a C-curve, 0 = K,W< KQ $ K,.
and spirals whose curvature runs from 0 to Ko), and contains the single spiral region of Fig. 5. Case 2. Consider the case when the minimum curvature is positive, 0
-n(O))
+.G@)
-n(llr>)+
(3.3)
where 0 Q s < S = l/K, - l/K,, 0 < G G 8. Now, considering the spiral from M to Q and using (2.2) (rotate the region (2.2) by 8 and translate its origin to M), then Q can be in the interior of the region
w&w-~)-n(O))
M(s, 8,i,b) +R(O)
-t(n(w--0)
--n(P))
(3.4)
M
where 0 d t G T= l/K, sion for this region is
- l/K,,
L(s, t, 8, cp, ct) =
0 G 50< W- B, 0 G 8 Q W. A simplified form of the exPres-
+r(W)- $n(O) - Sn( 0) P
.M
+s(n(0> -n($,))
-+(W>
-4e+cp1>,
(3.5)
whereO~s~~,O~tgT,O~~~~,O~~~W-6,O~~gW.Noticethefollowingproperties of L(s, I, 6, q, $1: if s = 0, Ji is irrelevant; if I = 0, p is irreievant; if 8 = $, s is irrelevant so one can set s = 0; and if W = 8 + 9, t is irrelevant, so one can set t = 0. Henceforth, the parameters of L will be listed with the first two set to zero whenever possible. With five parameters to think about, it is rather difficult to see what region is described by L. Consider first the parameters s and t. As they vary, a parallelogram is covered (Fig. 8). The vertices of that parallelogram are as follows: A=A(B)=L(O,0,6,
*, *),
(3.6)
B =B(@, 9) =L(O, T, 8, ~3, *>,
(3.7)
C = C(@, 9, $) = L(S, 2-t 8,40, tlr),
(3.8)
D.S. Meek. R.S.D. Thomas / Pau of spirals
198
Fig. 8. Parallelogram
associated
with a C-curve
with
and D=D($)
To facilitate
the labelling
E=L(O, H=H(e,
Notice
8, *, (cl) =L(O,O,
=L(s,O,
CL, *, *).
(3.9)
of Figs. 8, 9 and 10, let E, F, G, If, Z, J be defined as follows:
0, 0, *, *), ‘p) =L(S,
F=L(O,
T,O,O,
*),
T, e, cp, O),
z=~(e)
=H(e,
0) =L(S,
J=J(e)
=L(O,
T, 0,0, *).
G=L(O,O,
W
*, *>.
(3.10) (3.11) (3.12)
T, e, 0, O),
(3.13)
that H(B,
is independent qe,
W-
8) =E
of 8 and qc)
-H(e,
cp) =A(B)
--E=qn(O)
-n(e))
(3.14)
is independent of cp. As $ varies, 0 < II, < 0, each end of the line segment DC moves from E and H at I,/I= 0 by (3.10) and (3.11) to A and B at $ = 0 in a circular arc with radius S. Every point of the arc ADE outside the line segment AE is covered for some choice of $. Every point of the arc BCH lies inside the line segment BH and so is covered by the one choice $ = 0. Hence, the whole region EABH, with EH, AB, BH straight and EDA a circular arc, is covered (Fig. 9). As cp varies, 0 < cp < W - 8, H moves from Z = H(e, 0) to E = H(e, W - 0) in a circular arc of radius T while BH remains parallel and equal to AE as a result of B’s motion. (3.14). The whole region EAJI is covered, AJ, JI straight lines and, EA, IE circular arcs (Fig. 10). A short calculation shows for 0 < B < W that F, I(e) and J(0) are collinear, and that il(f?>, G and J(0) are collinear.
499
D.S. Meek, R.S.D. Thomas / Pair of spirals
Fig. 9. The region
EABH associated
with a C-curve
with 0 < K,,,.
As 8 varies, 0 G 0 Q W, Z moves from F = Z(O) to E = Z(W) in the circular arc of radius T on which H moves between Z and E, and A moves from E to C in the circular arc of radius S on which D moves between A and E. This covers the region shown in Fig. 11. The boundaries of the region are US, T, 0, 0, 0) and L(0, 0, 6, *, *I with 0 G B G W, and Lb, 0, W, *, 0) with 0 G s < S - T.
Fig. 10. The region
EA.Jl associated
with a C-curve
with 0
< K,,.
500
D.S. Meek, R.S.D. Thomas
/ Pair of spirals
Fig. 11. The region associated with a C-curve with 0 < K, < Kg $
KP.
As K, approaches zero, the circular arcs Gs~_Eand FZE approach respectively their common tangent ray at G and the horizontal common tangent ray at F, the boundaries when K.,, = 0 (Fig. 7). One wants the degrees of freedom ahowed by the choices of s, t, 0, cp and $, but the region is covered if one lets s = t move along AJZ and makes AH = .4Z by letting rp = II,= 0 and uses 8 to move Al.
4. U-curve
The U-curve considered here is a G* curve made by joining a pair of spirals of nonnegative curvature so that the curvature of the U-curve increases from K, at P to a maximum value, K,,, then decreases to K, at Q. Either end of a U-curve can be considered the start; so, without loss of generality, assume that K, ,
(4.1)
D.S. Meek. R.S. D. Thomas / Pair of spirals
501
Fig. 12. Region associated with a I/-cunx, 0 = K, = K,.
considering the spiral from M to Q and using (2.3) (2.3) by 0 and translate its origin to MI, then Q can be in the interior of the
where
s > 0, x a 0, 0 Q 13( W. Now,
(rotate
the region
region
e>+l?(t3)
M(s, x,
i
&(W-e)
Ti
-n(O))
fy
ii4
(( n
z
f 2 0, y a 0, 0 G 0 G
A =
&(W)
w-e
1)
e) -n(o))
+t(n(wwhere
+
1 ,
W. This expression can be thought of as a point
(4.2)
-n(O))
M
plus the convex sum of four vectors +)in( 4 + W) -t+(e)
Xn( 4)
--n(o))
+t(qv)
--n(e)),
(4.3)
where s 2 0, t > 0, x 2 0, y 2 0, 0 G 8 G W. The first vector is parallel to the x-axis, the second vector forms an angle of W with the x-axis, the third vector forms an angle of e/2 with the x-axis, and the fourth vector forms an angle of (0 + W)/2 with the x-axis. Since 0 < 0 < W, the third and fourth vectors are between the first and second and are redundant; without loss of generality set s = t = 0. The region is a wedge shape with apex A and bounded by lines parallel to the x-axis and at angle W. The boundaries of the region are obtained from (4.2) and (4.3) when s = t = y = 0; and when s = r =x = 0 (Fig. 12). Case 2. Consider the case when the curvatures satisfy 0 = K,
X,
e) +R(e)
$(t2(w-e) i
.Cf
-n(o))
+t(tqw-6)
-n(+))
,
502
D.S. Meek. R.S. D. Thomas / Pair of spirals
Fig. 13. The truncated wedge associated with a U-curve with 0 = K, -c K,.
where s 2 0, 0 Q t < T = l/K,
- 1/K,,,
0 G rc/< W - 8, 0 G 8 G W. This expression can be
thought of as a point
(4.4) plus a convex sum of three vectors xn
i
5 1+s(n(B)
-n(O))
t-t(n(W>
-n(cp)),
(4.5)
where s z 0,O G t G T, x a 0, 0 G p = 8 + $ zg W, 0 < 8 G W. The first vector is parallel to the x-axis, the second vector forms an angle of 8/2 with the x-axis, the third vector forms an angle f W + 9)/2 with the x-axis, but its length is restricted. These three vectors generate a truncated wedge (Fig. 13). The finite side of the wedge is a result of the restricted range of t, and the two end points of that side, at t = 0 and t = T, are A and
As p varies from II to W, B(p) moves along a circular arc from C = B(B) to A = B(W). The finite side of the wedge from A to B makes an angle of (W+ qD)/2 with the x-axis; the vector from B to C forms an angle of (p/2 with the x-axis, cp2 13.This means that the upper unbounded side of the truncated wedge from B(q), at angle 0/2, passes through or below C. If $ = 0, so that ~7= 8, then the ray s(n(8) -n(O)) passes through C and it is plain that the truncated wedges cover the region shown in Fig. 14. The angle 4 need not be set to zero in the formation of curves; this step mereIy clarifies the region’s boundaries, which are obtained from(4.4)and(4.5)whens=t=O,x~O;whens~T,t=x=0,8=W,andwhens=x=0, r=T:O<(p&W.
D.S. Meek, R.S.D. Thomas / Paw of spirals
Fig. 14. The region associated
with a U-curve with 0 = K, < KQ,
Case 3. The U-curve case with 0
K,v > 0. Assuming 0 < K, Q K,,
0 < Cc,< 8, 0 < B < W. Now, considering the spiral from M whereO
$-(n(W--ej
+)+23(e) i
where 0 < t < T = l/K, - l/K, be found using an analysis similar region are obtained from (4.6) OGSQS-T, Ijl=O, e= W(Fig.
,M
-n(o))
+t(n(w-e)
-n(q))
,
(4.6)
1
and 0 G cpQ W - 8. Simpler expressions for this region can to the one for Case 2 of the C-curve. The boundaries of the when s = t = 0; when s = S, t = T, $ = cp = 0; and when 15).
5. S-curve
The S-curve considered here is a G* curve made by joining a pair of spirals at a point of zero curvature M so that the curvature decreases from K, > 0 at P to K, < 0 at Q. Assume that the magnitudes of the angles through which the unit tangent vector rotates in PM and in QM are each less than r. The spiral whose unit tangent vector rotates through the larger angle can be taken first, so without loss of generality, assume W a 0. Let the angle that the
504
D.S. *Week, R.S.D. Thomas / Pair of spirals
- n(o))
Fig. 15. The region associated with a U-curve with 0 < KP -$ KQ.
unit tangent vector at M makes with the x-axis be 8, IV< B < T. The spiral from P to M, by f2.3), has M(s, X, 0) in the interior of the region
AWN-49)
(Ti )
+xn -f +e +%+2(e) -n(O)),
where .V3 0, x 2 0. Now, in considering the spiral from M to Q, a version of (2.1) where the y-coordinates are negated must be used. Note that negating the y-coordinate of n(0) gives -n( - 8). Rotate the modified region (2.1) by 8 and translate its origin to M to see that Q can be in the interior of the region
M(S, X, 0) +v)
i
z(
-+v-- 8) -h(o)) -t-t(
yn(- ;)
-
-qv-
e)
+ n(o))
i
,
where t & 0, y 2 0, W G 8 G T. This expression can be thought of as a point
.4(e) =
-L(w)- m(o)1P KQ
+
( +
P
-L n(e)? KQ
1
(5-l)
plus the convex sum of three vectors
(x+)+2(:+8)
t-+(e)
-n(o))
+t(n(e)
-n(w)),
(5.2)
where s 3 0, t > 0, x 2 0, y I+ 0 and WG B G ?r. The first vector forms an angle of 8 with the x-axis, the second vector forms an angle of 8/Z with the x-axis, and the third vector forms an angle of (W+ @I/2 with the x-axis. Since W< 8 < lip, the third vector is redundant as it is
D.S. Meek. R.S.D. Thomas / Pair
ofspirals
angle Q
Fig. 16. Wedge associated with an S-curve.
between the other two vectors: without loss of generality set t = 0. For each 8, the region is a wedge shape with apex A(B) and bounded by lines that form angles e/2 and B with the x-axis (Fig. 16). As A(8) moves around its circular path from B =A(W) to C =A(rr), the left arm of the wedge is always tangent to the circular path (recall that n(e> forms an angle of 8 - ~i/2 with the x-axis). Thus, it is obvious that the wedge sweeps out the region shown in Fig. 17. The boundaries of the region are obtained from (5.1) and (5.2) when x = y = 0, 8 = CV;when s-t ==x=y=O; andwhen s=t=O, 9=~. This region extends and includes the single spiral region with K, > 0 = K, (Fig. 4), being in fact the case Kp > 0 > K,.
6. An example Suppose a C-shaped curve consisting of two spirals with minimum curvature K, required between P and Q where the curvatures at those points are K, and K,, 0
Fig. 17. Region associated with an S-curve.
is <
506
D.S. Meek, R.S.D. Thomas / Pair of spirals
P W
Q
curvature K,
PJ
curvature K,
Fig. 18. A C-shaped pair of spirals joining P to Q.
K, 4 K,, and the angle between the unit tangent vectors rp and tQ is 0 < W< r (see Fig. 18). The problem is simplified by finding a point M, cuffature K,, at which the two spirals meet. The point Q must be in the region illustrated in Fig. 11; thus, Q can be expressed as L(s,t,tl,cp,q!t), equation (3.5), where O. Increase J&(Fig. 8) so that the parallelogram ABCD contains the point Q. Once Q is in the paralleIogram, the values of s and t are the coordinates of Q expressed in an oblique coordinate system using the sides of the paraIleIogram as axes. With vaiues for the five parameters s, t, 8, p and $ known, a valid point M is given by equation (3.31, and the angle that the unit tangent vector at M forms with the x-axis is 6. Note that M and 6 are not unique. The example can be further specialized to find a C-shaped clothoid spline that joins P to Q matching curvatures and unit tangent vectors at P and Q. Clothoids are spirals defined in terms of Fresnel integrals [Heald ‘8.51which have the property that their curvature varies linearly with respect to arc length [Nutbourne et al. ‘72, Stoer ‘821. Clothoid splines are G* curves consisting of clothoids, straight line segments and circular arcs. While they are not the only spirals used in practice, clothoid spiines have traditionally been used as the centre iines of highway and railway routes [Baass ‘841. An important paper concerning Hermite interpolation with clothoid splines constructed from up to four spiral segments is [Pal & Nutbourne ‘771. A C-shaped clothoid spline made of two clothoid spline spirals could now be found to join P to Q matching curvatures and unit tangent vectors at P and Q. If a point M is chosen as above in this section, the algorithm in [Meek & Walton ‘921 can be used to find a clothoid spline spiral from P to M, and another dothoid spiine spiral from M to Q. These two dothoid spline spirals join to form the required C-shaped clothoid spline. An interactive technique similar to the above can be used to find the other types of C-shaped clothoid splines, the U-shaped clothoid splines, and the S-shaped clothoid splines.
The authors acknowIedge the hefpful comments from two anonymous referees.
D.S. Meek. R.S. D. Thomas / Pair of spirals
507
References Baass, K.G. (198-t). The use of clothoid templates in highway design. Transportation Forum 1. -17-52. BCzier, P. (1972). ,Vumerical Control Mafhemarics and Applications, translation by A.R. Forrest and A.F. Frankhurst. Wiley, London. de Boor, C., Hollig, K. and Sabin, M. (1987), High accuracy geometric Hermite interpolation, Computer Aided Geometric Design 4, 269-278. Goodman, T.N.T. and Unsworth, K. (1988). Shape preserving interpolation by curvature continuous parametric cubits, Computer Aided Geometric Design 5, 323-340. New York. Guggenheimer, H.W. (1963). Diff erenrial Geometry. McGraw-Hill, Heald, M.A. (1985). Rational approximations for the Fresnel integrals, Math. Comput. 44, 459-461; 46, 771. Meek, D.S. and Walton. D.J. (1992), Clothoid spline transition spirals, Math. Comput. 59, 117-133. Nutbourne, A.W. and Martin, R.R. (19881, Diff erential Geometry Applied IO Curve and Surface Design, Volume I: Foundations, Ellis Hotwood, Chichester, UK. Nutbourne. A.W., McLellan, P.M., and Kensit, R.M.L. (1972), Curvature profiles for plane curves. Computer-Aided Design 4, 176-184. Pal, T.K., and Nutbourne, A.W. (1977), Two-dimensional curve synthesis using linear curvature elements, ComputerAided Design 9. 121-134. Pottmann, H. (1991), Locally controllable conic splines with curvature continuity, ACM Trans. Graphics 10, 366-377. Stoer, J. (1982), Curve fitting with clothoidal splines, J. Res. Nat. Bureau of Standards 87, 317-346. Su, B.Q. and Liu, D.Y. (19891, Computational Geometry-Curce and Surface Modeling, translated by G.-Z. Chang, Academic Press, San Diego. Wang, C.Y. (1981), Shape classification of the parametric cubic curve and parametric B-spline cubic curve. Computer-Aided Design 13, 199-206.
COMAID
Design
10 (1993) 491-507
491
322
Hermite interpolation of spirals *
with a pair
D.S. Meek Department of Computer Science, Unk,ersity of Manitoba, Winnipeg, Canada
R.S.D. Thomas St. John T Collrge and Department of Applied Mathematics, Unicersity of Manitoba. Winnipeg, Canada Received July 1991 Revised December 1992
Abstract Meek. D.S. and R.S.D. Thomas, Design 10 (1993) 491-507.
Hermite
interpolation
with a pair of spirals,
The Hermite interpolation problem in the plane considered tangent vectors and signed curvatures at the two points with The rotation of the tangent vector of the interpolating curve less than r. The necessary and sufficient conditions for the Keywords. Hermite
interpolation;
Computer
Aided
Geometric
here is to join two points and to match given unit various G2 curves consisting of a pair of spirals. from one point to the other is restricted to being existence of each of the various curves are given.
spirals.
1. Introduction
The unit tangent vector is a unit vector that is parallel to the tangent line and that points in the direction of increasing arc length. Curvature is a signed quantity which is positive if the centre of curvature is on the left when moving in the direction of increasing arc length. A spiral is a curve with monotone curvature of constant sign. A G* curve is a continuous curve with continuous unit tangent vector and continuous curvature. The Hermite interpolation problem is to find a curve which passes through given points and also matches higher-derivative data such as tangents and curvatures at those points. A biarc, a pair of circular arcs joined with matching tangent, can be used as a Hermite element that passes through two given points and matches unit tangents at the two points [Nutbourne & Martin ‘88, Su &Liu ‘891. A curve made of up to four clothoids can be used as a G* Hermite element that matches unit tangent vectors and curvatures at its endpoints [Pal & Nutbourne ‘771. A pair of conic segments can also be used as a G2 Hermite element that matches unit tangents and curvatures at its endpoints [Pottmann ‘911. The planar Her-mite interpolation problem considered here is to join two given distinct points P and Q, to match given unit tangent vectors, t,, tQ, and signed curvatures K,, K,, at Correspondence to: D. Meek, Department of Computer Science, University of Manitoba, Winnipeg (Man.). Canada R3T 2N2. Email addresses of the authors, Dereck_ [email protected] and Robert [email protected]. * This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada. 0167~8396/93/506.00
D 1993 - Elsevier
Science
Publishers
B.V. All rights reserved
192
D.S. .Cleek. R.S.D. Thomas / Pair of spirals
Fig. 1. Data in standard
form with radii of curnature
indicated.
P and Q with various G’ curves consisting of a pair of spirals. The new results presented here generalize Guggenheimer’s results for interpolation with a single spiral [Guggenheimer ‘631.A standard form of the interpolation problem is obtained by placing P at the origin, aligning t, along the positive r-a.xis, and reflecting across the x-axis if necessary so that K, 2 0 (Fig. 1). Let the rotation of the tangent vector from t, to tQ be W. The angle W is restricted, 0 < W < T, because the results for a spiral in [Guggenheimer ‘631 break into two cases depending on whether the angle is less than or greater than r. The restriction on W is sensible as one would not normally want the curve to rotate too much between two interpolation points, and it means that the curves will not self-intersect or form loops. The necessary and sufficient conditions for the existence of a single spiral joining P to Q and matching t,, tQ, K,, K, have been studied in [Guggenheimer ‘631 and [Meek & Walton ‘921. The conditions are summarized in Section 2 of this paper. Of the possible G2 curves consisting of a pair of spirals, the following three GZ curves in standard form are the only ones that need be considered: a C-curve whose curvature decreases monotonically to a nonnegative minimum and then increases monotonically; a U-curve whose curvature increases monotonically to some given maximum and then decreases monotonically; and an S-curve whose curvature decreases monotonically and becomes negative. The above curves will be discussed in Sections 3-5. The reason for studying curves made of a pair of spirals is that such curves have considerable flexibiIi~, but at the same time are somewhat restrictive. For example, a C-curve or a U-curve has exactly one curvature extremum and no inflection points, while an S-curve has exactiy one inflection point. Spirals cannot self intersect or form cusps; commoniy used parametric cubic curves can [Wang ‘811. Spirals can be bounded by circular arcs since the spiral from any point is entirely inside or entirely outside the circle of curvature at that point [Guggenheimer ‘63, p.481 With each of the curves made from a pair of spirals, there exist choices of Q, W, K,, and K, such that there is no curve of that type joining P to Q. For example, it is impossibie to find a C-curve of positive curvature (and the rotation of the tangent vector restricted to less than r;) joining P to Q if Q is below the x-axis. Given W, K,, K,, a region of the plane will be described such that the curve exists if and only if Q is in the interior of that region. If it seems strange that an interpolation curve may not exist, recall that interpolation between two points matching unit tangent vectors and signed curvatures at the two points is not always possible even with parametric cubic curves [BCzier ‘72 (p. 1231, de Boor et al. ‘87, Goodman & Unsworth ‘881. For example, if the curvatures at the two end points P and Q of a cubic BCzier curve are zero, the Bezier curve is a straight line joining P to Q, and the tangents are determined. Thus, these tangents may not match given unit tangent vectors at P and Q.
D.S. kiwk. R.S.D. Thomas / Pair of spirals
493
2. Single spiral
The problem of using a single spiral to join two points. matching curvatures and unit tangent vectors at those points, is discussed in [Guggenheimer ‘631. The following is a summary of Guggenheimer’s results and those in [Meek Rr Walton ‘921. Assume that the spiral is not a straight line segment or circular arc, or that K, f K,. There are four cases: nondecreasing curvature starting with (a) zero and with (b) positive curvature, and nonincreasing curvature ending with (c) zero and with (d) positive curvature. Case (a). Consider spirals with nondecreasing curvature and starting with zero curvature, or 0 = K,
-40))
fxn
+.+2(W) (J
-n(O)),
(2.1)
where
and s > 0, x > 0. The vector n(e) forms an angle of 0 - 7r/2 with the positive x-axis. The expression (2.1) can be thought of as a point plus the convex sum of two vectors. Thus, the region described is a wedge-shaped region with apex at the point (l/KQ)(n(W) -n(O)) and boundaries obtained when s = 0 and x = 0 in (2.1) (Fig. 2). Case (b) Consider spirals with nondecreasing curvature and starting with positive curvature, 0
Fig. 2. Region
associated
with a spiral with 0 = K, < Kp.
D.S. Meek. R.S. D. Thomas / Pair of spirals
Fig. 3. Region
associated
with a spiral with 0 < K, < KQ.
Case (c) Consider spirals with nonincreasing curvature. If the spiral ends with zero curvature, 0 = K,
&w P
(
+w -40)) +xn -f?T
) +s(n(Jv)
-n(O)),
(2.3)
where s 2 0, x 2 0. The above expression can be thought of as a point plus the convex sum of two vectors. Thus, the region is a wedge-shaped region with apex at the point (l/K,)(n( W) n(O)) and boundaries obtained when x = 0 and s = 0 in (2.3) (Fig. 4).
Fig. 4. Region
associated
with a spiral with 0 < Kp < K,.
D.S. Meek, R.S. D. Thomas / Pair of spirals
Fig. 5. Region
associated
with a spiral with 0 < K, < K,.
Case (d) Consider spirals with nonincreasing curvature and ending with positive 0
curvature, and unit
(2.4) where 0
3. c-curve The C-curve considered here is a G’ curve made by joining a pair of spirals of nonnegative curvature so that the curvature of the C-curve decreases from a positive K, at P to a minimum value, K,, then increases to a positive K, at Q. Either end of the C-curve could be taken as the starting point; so, without loss of generality, assume K, a K,. Let M be a point of minimum curvature, and let the angle that the unit tangent vector at M makes with the x-axis be 8, 0 < 8 < W. The analysis depends on whether K, is zero or positive. The region for K, = 0 includes the region for K, > 0,and is its limit as K,${ is allowed to approach zero. The case K, = 0 is treated first because it is simpler. Case 1. First, consider the case when the minimum curvature is 0, 0 = spiral from P to M, by (2.31, has M(s, x, 0) in the interior of the region
~0 P
T+e -40)) +xn (r
) +.+(e)-n(o)),
K, < KQ G K,.The
~2=0,~20.
496
D.S. Meek. R.S.D. Thomas / Pair of spirals
Fig. 6. Wedge
associated
with a C-curve,
K,, = 0.
Now, considering the spiral from M to Q and using (2.1) (rotate the region (2.1) by 0 and translate its origin to MI, then Q can be in the interior of the region M(s, x, S) +R(e)
i
&w-e)
-n(O))
+yn ;
KQ
i
j
+t(n(w-e)
-n(o))
where t 2 0, y a 0, 0 G 13d W, and R(B) is the rotation matrix R(B)
=
-sin “). cos I3
(?,”
The expression for this region can be thought of as a point
(3.1) where S = l/K,
- l/K,, ,?r
plus the convex sum of three vectors \
(3.2) where s > 0, t a 0, x 2 0, y a 0, 0 Q 0 Q W. The first vector forms an angle of 8 with the x-axis, the second vector forms an angle of e/2 with the x-axis, and the third vector forms an angle of (0 + WI/2 with the x-axis. Since 0 Q 8 < W, the first vector is redundant as it is between the other two vectors; without loss of generality, set x = y = 0. Thus, for each 8, the region is the interior of a wedge shape with apex A(B) and bounded by lines that form angles f3/2 and (0 + W)/2 with the x-axis (Fig. 6). As A(8) moves around its circular path from F =A(O) to G =A(W), the arm of the wedge at angle 8/2 always passes through F, and the extended arm at angle (0 + W)/2 always passes through G. Thus, it is obvious that the wedge sweeps out the nonconvex region shown in Fig. 7. The boundaries of the region are obtained from (3.1) and (3.2) with x = y = 0: the ray at angle W when s = 0 and 8 = W, the line segment at angle W/2 when 0
D.S. Meek. R.S. D. Thomas
/ Pair o/spirals
Fig. 7. Region associated with a C-curve, 0 = K,W< KQ $ K,.
and spirals whose curvature runs from 0 to Ko), and contains the single spiral region of Fig. 5. Case 2. Consider the case when the minimum curvature is positive, 0
-n(O))
+.G@)
-n(llr>)+
(3.3)
where 0 Q s < S = l/K, - l/K,, 0 < G G 8. Now, considering the spiral from M to Q and using (2.2) (rotate the region (2.2) by 8 and translate its origin to M), then Q can be in the interior of the region
w&w-~)-n(O))
M(s, 8,i,b) +R(O)
-t(n(w--0)
--n(P))
(3.4)
M
where 0 d t G T= l/K, sion for this region is
- l/K,,
L(s, t, 8, cp, ct) =
0 G 50< W- B, 0 G 8 Q W. A simplified form of the exPres-
+r(W)- $n(O) - Sn( 0) P
.M
+s(n(0> -n($,))
-+(W>
-4e+cp1>,
(3.5)
whereO~s~~,O~tgT,O~~~~,O~~~W-6,O~~gW.Noticethefollowingproperties of L(s, I, 6, q, $1: if s = 0, Ji is irrelevant; if I = 0, p is irreievant; if 8 = $, s is irrelevant so one can set s = 0; and if W = 8 + 9, t is irrelevant, so one can set t = 0. Henceforth, the parameters of L will be listed with the first two set to zero whenever possible. With five parameters to think about, it is rather difficult to see what region is described by L. Consider first the parameters s and t. As they vary, a parallelogram is covered (Fig. 8). The vertices of that parallelogram are as follows: A=A(B)=L(O,0,6,
*, *),
(3.6)
B =B(@, 9) =L(O, T, 8, ~3, *>,
(3.7)
C = C(@, 9, $) = L(S, 2-t 8,40, tlr),
(3.8)
D.S. Meek. R.S.D. Thomas / Pau of spirals
198
Fig. 8. Parallelogram
associated
with a C-curve
with
and D=D($)
To facilitate
the labelling
E=L(O, H=H(e,
Notice
8, *, (cl) =L(O,O,
=L(s,O,
CL, *, *).
(3.9)
of Figs. 8, 9 and 10, let E, F, G, If, Z, J be defined as follows:
0, 0, *, *), ‘p) =L(S,
F=L(O,
T,O,O,
*),
T, e, cp, O),
z=~(e)
=H(e,
0) =L(S,
J=J(e)
=L(O,
T, 0,0, *).
G=L(O,O,
W
*, *>.
(3.10) (3.11) (3.12)
T, e, 0, O),
(3.13)
that H(B,
is independent qe,
W-
8) =E
of 8 and qc)
-H(e,
cp) =A(B)
--E=qn(O)
-n(e))
(3.14)
is independent of cp. As $ varies, 0 < II, < 0, each end of the line segment DC moves from E and H at I,/I= 0 by (3.10) and (3.11) to A and B at $ = 0 in a circular arc with radius S. Every point of the arc ADE outside the line segment AE is covered for some choice of $. Every point of the arc BCH lies inside the line segment BH and so is covered by the one choice $ = 0. Hence, the whole region EABH, with EH, AB, BH straight and EDA a circular arc, is covered (Fig. 9). As cp varies, 0 < cp < W - 8, H moves from Z = H(e, 0) to E = H(e, W - 0) in a circular arc of radius T while BH remains parallel and equal to AE as a result of B’s motion. (3.14). The whole region EAJI is covered, AJ, JI straight lines and, EA, IE circular arcs (Fig. 10). A short calculation shows for 0 < B < W that F, I(e) and J(0) are collinear, and that il(f?>, G and J(0) are collinear.
499
D.S. Meek, R.S.D. Thomas / Pair of spirals
Fig. 9. The region
EABH associated
with a C-curve
with 0 < K,,,.
As 8 varies, 0 G 0 Q W, Z moves from F = Z(O) to E = Z(W) in the circular arc of radius T on which H moves between Z and E, and A moves from E to C in the circular arc of radius S on which D moves between A and E. This covers the region shown in Fig. 11. The boundaries of the region are US, T, 0, 0, 0) and L(0, 0, 6, *, *I with 0 G B G W, and Lb, 0, W, *, 0) with 0 G s < S - T.
Fig. 10. The region
EA.Jl associated
with a C-curve
with 0
< K,,.
500
D.S. Meek, R.S.D. Thomas
/ Pair of spirals
Fig. 11. The region associated with a C-curve with 0 < K, < Kg $
KP.
As K, approaches zero, the circular arcs Gs~_Eand FZE approach respectively their common tangent ray at G and the horizontal common tangent ray at F, the boundaries when K.,, = 0 (Fig. 7). One wants the degrees of freedom ahowed by the choices of s, t, 0, cp and $, but the region is covered if one lets s = t move along AJZ and makes AH = .4Z by letting rp = II,= 0 and uses 8 to move Al.
4. U-curve
The U-curve considered here is a G* curve made by joining a pair of spirals of nonnegative curvature so that the curvature of the U-curve increases from K, at P to a maximum value, K,,, then decreases to K, at Q. Either end of a U-curve can be considered the start; so, without loss of generality, assume that K, ,
(4.1)
D.S. Meek. R.S. D. Thomas / Pair of spirals
501
Fig. 12. Region associated with a I/-cunx, 0 = K, = K,.
considering the spiral from M to Q and using (2.3) (2.3) by 0 and translate its origin to MI, then Q can be in the interior of the
where
s > 0, x a 0, 0 Q 13( W. Now,
(rotate
the region
region
e>+l?(t3)
M(s, x,
i
&(W-e)
Ti
-n(O))
fy
ii4
(( n
z
f 2 0, y a 0, 0 G 0 G
A =
&(W)
w-e
1)
e) -n(o))
+t(n(wwhere
+
1 ,
W. This expression can be thought of as a point
(4.2)
-n(O))
M
plus the convex sum of four vectors +)in( 4 + W) -t+(e)
Xn( 4)
--n(o))
+t(qv)
--n(e)),
(4.3)
where s 2 0, t > 0, x 2 0, y 2 0, 0 G 8 G W. The first vector is parallel to the x-axis, the second vector forms an angle of W with the x-axis, the third vector forms an angle of e/2 with the x-axis, and the fourth vector forms an angle of (0 + W)/2 with the x-axis. Since 0 < 0 < W, the third and fourth vectors are between the first and second and are redundant; without loss of generality set s = t = 0. The region is a wedge shape with apex A and bounded by lines parallel to the x-axis and at angle W. The boundaries of the region are obtained from (4.2) and (4.3) when s = t = y = 0; and when s = r =x = 0 (Fig. 12). Case 2. Consider the case when the curvatures satisfy 0 = K,
X,
e) +R(e)
$(t2(w-e) i
.Cf
-n(o))
+t(tqw-6)
-n(+))
,
502
D.S. Meek. R.S. D. Thomas / Pair of spirals
Fig. 13. The truncated wedge associated with a U-curve with 0 = K, -c K,.
where s 2 0, 0 Q t < T = l/K,
- 1/K,,,
0 G rc/< W - 8, 0 G 8 G W. This expression can be
thought of as a point
(4.4) plus a convex sum of three vectors xn
i
5 1+s(n(B)
-n(O))
t-t(n(W>
-n(cp)),
(4.5)
where s z 0,O G t G T, x a 0, 0 G p = 8 + $ zg W, 0 < 8 G W. The first vector is parallel to the x-axis, the second vector forms an angle of 8/2 with the x-axis, the third vector forms an angle f W + 9)/2 with the x-axis, but its length is restricted. These three vectors generate a truncated wedge (Fig. 13). The finite side of the wedge is a result of the restricted range of t, and the two end points of that side, at t = 0 and t = T, are A and
As p varies from II to W, B(p) moves along a circular arc from C = B(B) to A = B(W). The finite side of the wedge from A to B makes an angle of (W+ qD)/2 with the x-axis; the vector from B to C forms an angle of (p/2 with the x-axis, cp2 13.This means that the upper unbounded side of the truncated wedge from B(q), at angle 0/2, passes through or below C. If $ = 0, so that ~7= 8, then the ray s(n(8) -n(O)) passes through C and it is plain that the truncated wedges cover the region shown in Fig. 14. The angle 4 need not be set to zero in the formation of curves; this step mereIy clarifies the region’s boundaries, which are obtained from(4.4)and(4.5)whens=t=O,x~O;whens~T,t=x=0,8=W,andwhens=x=0, r=T:O<(p&W.
D.S. Meek, R.S.D. Thomas / Paw of spirals
Fig. 14. The region associated
with a U-curve with 0 = K, < KQ,
Case 3. The U-curve case with 0
K,v > 0. Assuming 0 < K, Q K,,
0 < Cc,< 8, 0 < B < W. Now, considering the spiral from M whereO
$-(n(W--ej
+)+23(e) i
where 0 < t < T = l/K, - l/K, be found using an analysis similar region are obtained from (4.6) OGSQS-T, Ijl=O, e= W(Fig.
,M
-n(o))
+t(n(w-e)
-n(q))
,
(4.6)
1
and 0 G cpQ W - 8. Simpler expressions for this region can to the one for Case 2 of the C-curve. The boundaries of the when s = t = 0; when s = S, t = T, $ = cp = 0; and when 15).
5. S-curve
The S-curve considered here is a G* curve made by joining a pair of spirals at a point of zero curvature M so that the curvature decreases from K, > 0 at P to K, < 0 at Q. Assume that the magnitudes of the angles through which the unit tangent vector rotates in PM and in QM are each less than r. The spiral whose unit tangent vector rotates through the larger angle can be taken first, so without loss of generality, assume W a 0. Let the angle that the
504
D.S. *Week, R.S.D. Thomas / Pair of spirals
- n(o))
Fig. 15. The region associated with a U-curve with 0 < KP -$ KQ.
unit tangent vector at M makes with the x-axis be 8, IV< B < T. The spiral from P to M, by f2.3), has M(s, X, 0) in the interior of the region
AWN-49)
(Ti )
+xn -f +e +%+2(e) -n(O)),
where .V3 0, x 2 0. Now, in considering the spiral from M to Q, a version of (2.1) where the y-coordinates are negated must be used. Note that negating the y-coordinate of n(0) gives -n( - 8). Rotate the modified region (2.1) by 8 and translate its origin to M to see that Q can be in the interior of the region
M(S, X, 0) +v)
i
z(
-+v-- 8) -h(o)) -t-t(
yn(- ;)
-
-qv-
e)
+ n(o))
i
,
where t & 0, y 2 0, W G 8 G T. This expression can be thought of as a point
.4(e) =
-L(w)- m(o)1P KQ
+
( +
P
-L n(e)? KQ
1
(5-l)
plus the convex sum of three vectors
(x+)+2(:+8)
t-+(e)
-n(o))
+t(n(e)
-n(w)),
(5.2)
where s 3 0, t > 0, x 2 0, y I+ 0 and WG B G ?r. The first vector forms an angle of 8 with the x-axis, the second vector forms an angle of 8/Z with the x-axis, and the third vector forms an angle of (W+ @I/2 with the x-axis. Since W< 8 < lip, the third vector is redundant as it is
D.S. Meek. R.S.D. Thomas / Pair
ofspirals
angle Q
Fig. 16. Wedge associated with an S-curve.
between the other two vectors: without loss of generality set t = 0. For each 8, the region is a wedge shape with apex A(B) and bounded by lines that form angles e/2 and B with the x-axis (Fig. 16). As A(8) moves around its circular path from B =A(W) to C =A(rr), the left arm of the wedge is always tangent to the circular path (recall that n(e> forms an angle of 8 - ~i/2 with the x-axis). Thus, it is obvious that the wedge sweeps out the region shown in Fig. 17. The boundaries of the region are obtained from (5.1) and (5.2) when x = y = 0, 8 = CV;when s-t ==x=y=O; andwhen s=t=O, 9=~. This region extends and includes the single spiral region with K, > 0 = K, (Fig. 4), being in fact the case Kp > 0 > K,.
6. An example Suppose a C-shaped curve consisting of two spirals with minimum curvature K, required between P and Q where the curvatures at those points are K, and K,, 0
Fig. 17. Region associated with an S-curve.
is <
506
D.S. Meek, R.S.D. Thomas / Pair of spirals
P W
Q
curvature K,
PJ
curvature K,
Fig. 18. A C-shaped pair of spirals joining P to Q.
K, 4 K,, and the angle between the unit tangent vectors rp and tQ is 0 < W< r (see Fig. 18). The problem is simplified by finding a point M, cuffature K,, at which the two spirals meet. The point Q must be in the region illustrated in Fig. 11; thus, Q can be expressed as L(s,t,tl,cp,q!t), equation (3.5), where O
The authors acknowIedge the hefpful comments from two anonymous referees.
D.S. Meek. R.S. D. Thomas / Pair of spirals
507
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