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Caustics of spline curves∗
Ede Troll, Miklós Hoffmann
Institute of Mathematics and Computer Science
Eszterházy Károly University, Eger, Hungary
[troll.ede,hoffmann.miklos]@uni-eszterhazy.hu
Submitted September 26, 2017 — Accepted December 18, 2017
Abstract
If we consider a curve as a mirror, then parallel light rays reflected by
this mirror curve form a family of lines, which generally has an envelope.
This envelope is called caustic curve of the given mirror curve. The aim of
this paper is to describe the caustic curve of a free-form curve and study its
alteration by the modification of a control point. We will provide results of
general form, and case studies in terms of Bézier and B-spline curves.
Keywords: caustic curve, Bézier curve, B-spline curve
1. Introduction
The classical definition of caustic curves can be found in several books (e.g. in
[1, 2]), which can be adapted for parameterical curves as follows.
Definition 1.1. Given a direction vector v(vx, vy)of light rays and the mirror
curve r(x(t), y(t)), t ∈[0,1], the envelope of rays (if exists) reflected by the curve
is called caustic curve (or simply caustic) of the given curve.
More precisely the curve defined above is called catacaustic curve, making dis-
tinction between reflected and refracted rays – in this latter case the envelope curve
is called diacaustic curve. Throughout the paper we will consider only reflected
rays, therefore the curve will simply be called caustic.
Although the study of caustic curves has a long history in mathematics and
optics, the shape of the mirror is generally a classical curve, such as line, circle
∗This research was supported by the grant EFOP-3.6.1-16-2016-00001 (Complex improvement
of research capacities and services at Eszterházy Károly University).
Annales Mathematicae et Informaticae
47 (2017) pp. 201–209
http://ami.uni-eszterhazy.hu
201
or parabola. In the case of reflection by a circle, the caustic is an epicycloid, the
locus of a fixed point on the circumference of a circle that rolls on the exterior of
another circle, whilst in the case of reflection by the parabola, the caustic is a cubic
algebraic curve called Tschirnhausen’s cubic (see e.g. [3]).
But there are limited results in geometric modeling considering caustic curves
of other curves, especially caustic curve of free-form curves. While it is relatively
easy to visualize the family of rays for any type of mirror curve, and the human
perception can ‘realize’ the envelope if the family is dense enough, which can also
been approximated numerically providing predefined shapes ([4] and references
therein), the exact computation of the caustic curve yielding closed form of these
curves is a challenging problem. Exact computation of the evolute of Bézier curves
using computer algebra software can be found in [5]. The aim of this paper is
to provide exact mathematical description of the caustic curve of free-form mirror
curves with arbitrary basis functions in a closed form.
2. The caustic of a free-form curve
In this section we will discuss the method of calculating the caustic of a control
point based free-form curve. Let the curve be given in the form
r(t) =
n
X
i=0
Ai(t)pi,
where t∈[0,1],A(t)iare the basis functions and piare the control points.
When we emit parallel light rays onto the curve with the direction vector
v(vx, vy)then these rays will form the family of straight lines
l(u, t) = r(t) + uv,
where u∈Ris the running parameter of the lines, while tis the family parameter,
coinciding the running parameter of the mirror curve.
To obtain the reflected rays, we have to mirror these incoming rays onto the
corresponding tangent lines
t(w, t) = r(t) + wr0(t),
where w∈R.
To define the reflected rays we translated the points of the curve at every twith
the vector vand mirrored them onto the tangent lines. The mirrored points are
mr(t) (xm(t), ym(t)), where
xm(t) = x(t)−2hv,nt(t)ixn(t),
ym(t) = y(t)−2hv,nt(t)iyn(t),
202 E. Troll, M. Hoffmann
Figure 1: The family of incoming light rays and the corresponding
tangents
where nt(t) (xn(t), yn(t)) are the unit normal vectors of the tangents and
xn(t) = y0(t)
qx02(t) + y02(t)
,
yn(t) = −x0(t)
qx02(t) + y02(t)
.
Now the direction vector of the reflected rays are vr(t) = mr(t)−r(t)and the
family of the reflected rays can be described as
lr(u, t) = r(t) + uvr(t),
where u∈Rand the coordinate functions are
xl(u, t) = x(t) + uxvr(t)
yl(u, t) = y(t) + uyvr(t).
Figure 2: The family of reflected light rays
Caustics of spline curves 203
To compute the envelope, that is the caustic curve, we have to solve the following
equation for u
∂xl
∂t
∂xl
∂u
∂yl
∂t
∂yl
∂u = 0,
where
∂xl
∂t = 2u
2vxy02(t)−vyx0(t)y0(t)(x0(t)x00(t) + y0(t)y00(t))
x02(t) + y02(t)2
−2vxy0(t)y00(t)−vy(x0(t)y00(t) + y0(t)x00(t))
x02(t) + y02(t)
+x0(t),
∂xl
∂u =vx−2y0(t) (vxy0(t)−vyx0(t))
x02(t) + y02(t),
∂yl
∂t = 2u
2vyx02(t)−vxx0(t)y0(t)(x0(t)x00(t) + y0(t)y00(t))
x02(t) + y02(t)2
−2vyx0(t)x00(t)−vx(x0(t)y00(t) + y0(t)x00(t))
x02(t) + y02(t)
+y0(t),
∂yl
∂u =vy−2x0(t) (vyx0(t)−vxy0(t))
x02(t) + y02(t).
Substituting these forms into the original equation
∂xl
∂u
∂yl
∂t −∂yl
∂u
∂xl
∂t = 0
after some calculation the equation can be simplified in the following form
2uv2
x(x0(t)y00(t)−y0(t)x00(t)) + v2
y(x0(t)y00(t)−y0(t)x00(t))
x02(t) + y02(t)
+
vyx03(t) + x0(t)y02(t)−vxy03(t) + y0(t)x02(t)
x02(t) + y02(t)= 0
and the determinant equation gives us u(t)what is the parameter uof the rays
expressed by the family parameter tas follows
u(t) = −1
2
vyx03(t) + x0(t)y02(t)−vxy03(t) + x02(t)y0(t)
v2
x+v2
y(x0(t)y00(t)−x00(t)y0(t)) .
Now we can express the envelope of the reflected rays c(t) (xc(t), yc(t)) as the
function of the parameter tof the original free-formed curve where
xc(t) = xl(u(t), t)
204 E. Troll, M. Hoffmann
=−1
2
(vyx0(t)−vxy0(t)) vxx02(t)+2vyx0(t)y0(t)−vxy02(t)
v2
x+v2
y(x0(t)y00(t)−x00(t)y0(t)) +x(t),
yc(t) = yl(u(t), t)
=1
2
(vyx0(t)−vxy0(t)) vyx02(t)−2vxx0(t)y0(t)−vyy02(t)
v2
x+v2
y(x0(t)y00(t)−x00(t)y0(t)) +y(t).
Figure 3: The caustic curve of a control point based free-formed
curve
In these computations we extensively used computer algebra softwares. As we
can observe from the equation above, the caustic curve of a control point based
free formed curve is rational, and its degree depends on the degree of the original
basis functions.
Example 2.1. The construction of the caustic curve described in the preceding
section is general, that is valid for every control point based free-form curve, and
it is independent of the degree and type of the basis functions. Here we provide
some case studies of the results for the most popular free-form curves.
In Figure 4 a Bézier curve of degree 4 can be seen in the same position but with
different incoming direction of light rays.
Since the caustic has the same parameter interval t∈[0,1] as the original curve,
the piecewisely computed caustics of connect B-Spline arcs will also be smoothly
connected automatically. In Figure 5 a uniform cubic B-Spline and its piecewisely
connected caustic curve can be seen. To show the arcs of the caustics we drew
them with different colours.
The last example shows a closed uniform cubic B-Spline and its caustics with
various light ray directions, where the control points are vertices of a square (Fig 6).
In the leftmost image the incoming rays are parallel to the yaxis, while in the
right image the rays are parallel to the xaxis. These caustics look very similar to
the nephroid what is the caustic of the circle. In the middle image the direction
vector of the incoming rays does not parallel to any of the axes and because the
cubic B-Spline can not describe an exact circle, the caustic is slightly distorted.
Caustics of spline curves 205
Figure 4: The caustic of a Bézier curve of degree 4 with different
light directions
Figure 5: The caustic curve of the uniform cubic B-spline curve
(different colours means different arcs of the caustic)
Figure 6: The caustic of the closed uniform cubic B-Spline curve
with different light ray directions
2.1. Geometric effect of the control points
In this section, our aim is to describe the alteration of the caustic curve by the
modification of a control point. This means that we would like to compute the path
206 E. Troll, M. Hoffmann
of a point c(t0), t0∈[0,1] of the caustic curve when a control point pi, i ∈0,1, . . . , n
moves along a straight line. To describe the effect of the modification we have to
express the caustic curve in a way that the running parameter tis considered to
be constant (t=t0), and the control point pi(xpi, ypi)is the parameter (more
precisely, the coordinates of the control point are the parameters) of the path. To
emphasize this fact the path of the point c(t0)is denoted by s(t0,pi). Let us
express the coordinate functions of the free-form curve and their derivatives as a
constant part plus the part depending on the coordinates of the moving control
points:
x(t0, xpi) = X
j6=i
Aj(t0)xpj+Ai(t0)xpi, y(t0, ypi) = X
j6=i
Aj(t0)ypj+Ai(t0)ypi,
x0(t0, xpi) = X
j6=i
A0
j(t0)xpj+A0
i(t0)xpi, y0(t0, ypi) = X
j6=i
A0
j(t0)ypj+A0
i(t0)ypi,
x00(t0, xpi) = X
j6=i
A00
j(t0)xpj+A00
i(t0)xpi, y00(t0, ypi) = X
j6=i
A00
j(t0)ypj+A00
i(t0)ypi,
where jruns from 0 to n.
Using the expressions above we can expand the coordinate functions of the
caustics in the form
xc(t0, xpi) = −1
2
(vyx0(t0, xpi)−vxy0(t0, xpi))
v2
x+v2
y(x0(t0, xpi)y00(t0, xpi)−x00(t0, xpi)y0(t0, xpi))
×vxx02(t0, xpi)+2vyx0(t0, xpi)y0(t0, xpi)−vxy02(t0, xpi)
v2
x+v2
y(x0(t0, xpi)y00(t0, xpi)−x00(t0, xpi)y0(t0, xpi))
+x(t0, xpi),
yc(t0, ypi) = 1
2
(vyx0(t0, ypi)−vxy0(t0, ypi))
v2
x+v2
y(x0(t0, ypi)y00(t0, ypi)−x00(t0, ypi)y0(t0, ypi))
×vyx02(t0, ypi)−2vxx0(t0, ypi)y0(t0, ypi)−vyy02(t0, ypi)
v2
x+v2
y(x0(t0, ypi)y00(t0, ypi)−x00(t0, ypi)y0(t0, ypi))
+y(t0, ypi).
By these expressions we can observe that the paths s(t0,pi)are rational curves
(see Fig. 7) and their coordinate functions sxand sycan be expressed as
sx(t0, xpi) = a1x3
pi+a2x2
pi+a3
a4xpi+a5
,
sy(t0, ypi) = b1y3
pi+b2y2
pi+b3
b4ypi+b5
,
where akand bk,(k∈ {1,2,...,5})are constants depending on the direction vector
of the incoming rays, the fixed value of the running parameter t0and the position
of the other control points.
Caustics of spline curves 207
Figure 7: The geometric effect of the alteration of the control point
p2on the caustic curve and paths of points of a cubic Bézier curve
when the control point moves along the straight line defined by the
direction vector d(0.22,0.98)
3. Conclusion and future research
Caustic curves of control point based free-form curves have been computed and
provided in closed form in this paper. The theoretical results hold for any free-
form curve, independently of its specific basis functions. Beside these results, some
examples of the most popular free-form curves are also considered. As an important
aspect of the design of free-form mirrors and their caustics we have seen, that the
alteration of a control point will evidently affect the shape of the caustic, and fixed
points of the caustic curve move along rational curves as one of the control points
is altered along a straight line.
These results may form the basis for the spatial extension of the computation of
caustics of free-form surfaces, which is an important aspect of geometric modeling
and free-form architecture as well. It can be the direction of our future research.
References
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[4] Schwartzburg, Y., Testuz, R., Tagliasacchi, A., Pauly, M., High-contrast
computational caustic design, ACM Transactions on Graphics (TOG), 33(4) (2014),
1–11.
208 E. Troll, M. Hoffmann
[5] Tsukada, T., Caustics on Spline Curves, Wolfram Demonstration Project http:
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Caustics of spline curves 209
