=Paper=
{{Paper
|id=Vol-2744/paper66
|storemode=property
|title=Caustics, Orthotomics, and Reflecting Curve with Source at an Infinity
|pdfUrl=https://ceur-ws.org/Vol-2744/paper66.pdf
|volume=Vol-2744
|authors=Aleksandr Dvoretskii
}}
==Caustics, Orthotomics, and Reflecting Curve with Source at an Infinity==
https://ceur-ws.org/Vol-2744/paper66.pdf
Caustics, Orthotomics, and Reflecting Curve with Source
at an Infinity*
Aleksandr Dvoretskii[0000-0002-1152-768X]
The Crimean Federal University named after V.I. Vernadsky, Simferopol, Russia
erces_crimea@mail.ru
Abstract. The properties of the first reflection of curves can be investigated by
their connections. It is giving us the possibility to construct new reflecting curves
and reflecting systems with the demand properties. The studied reflective prop-
erties of plane curves as sections of reflecting surfaces are preserved if these sec-
tions are generators of surfaces of revolution and rotative surfaces (the normal to
the section at a given point coincides with the normal to the surface at the same
point) and the source of the incident rays is in the same plane section.
Caustics, orthotomics, and reflective curves with a source at an infinity are
considered. For a given orthotomics, there is a one-parameter set of reflective
curves with a source at an infinity. To select one reflecting curve from this set it
is necessary to put straight line, perpendicular to the direction of the incident rays.
The shaping of a reflecting curve and its caustics according to its given ortho-
tomics, as well as the shaping of orthotomics according to a given reflective curve
with a source at an infinity, is studied, and algorithms for these shaping are pro-
posed. Equations are written for all studied curves.
Caustics, orthotomics, and reflective curves with a source at infinity are a
promising means of using the methods of geometric optics for modeling natural
objects in ergo design.
Keywords: Orthotomic, Caustic, Shaping of a Reflecting Curve, a Source at an
Infinity
1 Statement of the problem
When the source is located in the plane of the reflecting curve, the envelope of the
reflected rays can be found, which is called caustic. Singularities of caustics, systems
of rays and fronts were studied by Huygens even before the emergence of mathematical
analysis (1654). In the works of Cayley (1878), studies of the singularities of wave
fronts and caustics made significant progress [1]. Caustics for a circle with a light
source at point L are shown in Figure 1 and described in [2].
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
2 A. Dvoretskii.
Fig. 1. Caustic for a circle Fig. 2. Caustic and orthotomic W of an el-
lipse M with a point source L located out of
focus.
The envelope of the normals of a curve is known as its evolute (and can also be thought
of as the set of canters of curvature of the curve). Thus, focusing occurs on the evolute
of W, called the orthotomic of M relative to L. Because of this optical connection, the
evolute is also known as the focal set of W, or indeed the caustic, but we reserve this
term for a caustic by reflection. The orthotomic was introduced into the study of caus-
tics by Quetelet in 1826. Caustics themselves appear to go back as far as Tschirnhaus
(1682); a detailed study of special cases was carried out by Cayley [9] in 1856.
In the works of Bruce, Gibling, Gibson [3,4], caustics and orthotomics of an ellipse
are considered for a point source located out of focus. Figure 2 shows the reflection
from an ellipse M with a source L, located inside the ellipse.
The objective of this article is to study the relationship between caustics, orthotomics
and a reflective curve with a source at an infinity.
The elements included in this apparatus are: catacaustics n - a curve enveloping a
one-parameter set of reflected rays and at the same time being an evolute of an
orthotomic o; podder curve p, which is the locus of points equidistant from orthotomics
and the corresponding point L on line 1.
For a given orthotomics, there is a one-parameter set of reflective curves with a
source at an infinity. It is possible to select one reflecting curve from this set by
specifying a straight line 1, perpendicular to the direction of the incident rays S (see
Fig. 2). The theory of quasi-focal lines for reflections in three-dimensional space is
described in [6].
Caustics, Orthotomics, and Reflecting Curve with Source at an Infinity 3
Fig. 3. The reflecting apparatus with a source at an infinity
2 Formation of the reflecting curve and its caustic
according to its given orthotomic with a source at an
infinity
It is proposed to construct a reflecting curve according to its given orthotomic according
to the following algorithm:
1. A point 0 (x, y) is chosen on the orthotomic o and the tangent t is constructed in this
point.
2. From the point T as a center, which is the point of intersection of tangent t and
straight line 1, a circle with radius TN cuts a point L on straight line 1.
3. The point of intersection of the normal to the orthotomic at point O with the incident
ray, passing through point L is the point of the reflecting curve M (x, y).
Let
y = f (x) (1)
is the orthotomics equation о, for which we find the reflecting curve m, and as a
straight line that distinguishes one reflecting curve, we take the x axis (see Fig. 3). The
tangent to the curve o at the point O (x, y) is described by the equation
Y − y = f ' ( x)( X − x) , (2)
where x and y are current coordinates.
Coordinates of point T, cut off by tangent (2) on the x-axis are
f ' ( x) x − y
XT = ; YT = 0 . (3)
f ' ( x)
4 A. Dvoretskii.
The distance from point T to point N will be
d = ( X T − x) 2 + (YT − y ) (4)
and with (3) is
1
d = y 1+ . (5)
f ' ( x) 2
Then the equation of the incident ray perpendicular to the x-axis at point L will be:
f ' ( x) x − (1 + 1 + f ' ( x) 2 ) y
X L = XT − d = = XM . (6)
f ' ( x)
The equation of orthotomics normal is
1
Y−y=− ( X − x) . (7)
f ' ( x)
The coordinates of the point M of the reflecting curve are determined by the inter-
section of the normal to the orthotomics (7) and the incident ray (6) passing through
point L.
Taking into account (1) and (3) we get
f ' ( x) x − (1 + 1 + f ' ( x) 2 ) y
XM = , (8)
f ' ( x)
( f ' ( x) 2 − 1 − 1 + f ' ( x) 2 ) y .
YM = (9)
f ' ( x) 2
This is the parametric equation of the reflective curve.
As an example, consider finding a reflective curve for orthotomic in the form of a
circle centered on the y-axis and a straight line 1 coinciding with the x-axis. Let
x = R cos t
. (10)
y = R sin t + a
is the parametric equation the circle о.
Derivative of the equation (10) will be
y ' (t )
y ' ( x) = = −ctgt . (11)
x' (t )
Caustics, Orthotomics, and Reflecting Curve with Source at an Infinity 5
Fig. 4. Reflecting curve for the orthothomic in the form of circle o is parabola m.
Using formulas (10) and (11), we obtain the coordinates of the current point of the
reflecting curve m, depending on the parameter t. After excluding the parameter t, the
equation takes the form:
x 2 = 2(a − R) y + R 2 − a 2 . (12)
Equation (12) shows that the reflecting curve for circular orthotomic is a parabola
(see Fig. 4).
Table 1 shows the reflecting curves for orthotomics in the form of an ellipse, parab-
ola, and cycloid, and their caustics. Figure 5 shows the result of the formation of four
curves by the reflections with a source at an infinity. The calculations were performed
in the MathCad program.
Table 1. Connection of orthothomic and caustic
Orthothomic Drawing Evolute-caustic
Ellipse 2 2
x a 3 y b 3
2 2 + 2 2 =1
a −b a −b
6 A. Dvoretskii.
Parabola
27 py 2 = 8( x − p )
x = a + a (1 − sin t ' )
Cycloid
y = −2a + a(1 − cos t ' )
t' = t +
Fig. 5. Reflecting curves on the given orthothomic as a parabola.
Caustics, Orthotomics, and Reflecting Curve with Source at an Infinity 7
Table 2. Reflecting curves on the given orthothomic as a parabola
Reflecting curve Podera Othothomic Caustic
4 p x − x3
xp = − xk =
xkr = x − (b − y ) tg 4 p p2
x2
(2 b p − x ) tgx
2
y=
x 3 + p tg −
4 p
2 p 2 p2 + 3 x2
y kr = yk =
2 x p 2 p
2 b p + x2
(2 b p − x 2 ) yp =
4 p
3 Shaping of orthotomics according to a given reflective
curve with a source at an infinity
Since an orthotomic determines the reflective properties of the curve, it is advisable to
consider the shaping of orthotomics on a given reflective curve with a source at an
infinity.
Statement. For a given reflecting curve m y = f (x), orthotomics is one of two enve-
lopes of a set of circles of variable radius centered on the reflecting curve. Another
envelope is a straight line 1, which defines a beam of parallel incident rays and is per-
pendicular to these rays (Fig. 6).
Finding one of the envelopes, if the other is a straight line, is proposed using the
following algorithm:
1. The point M (x, y) on the reflecting curve m, y = f (x, y), is selected.
2. Construct a tangent t to the reflecting curve at the point M (x, y).
3. Find a point A on line 1, simultaneously belonging to the incident ray passing
through the point M (x, y).
4. Point O (Xo.Yo) of orthotomic is constructed as a point symmetric to point A relative
to the tangent t at the point M (x, y).
Let a reflecting curve in the form of a circle be given (Fig. 6)
( y − a)2 + x 2 = R 2 . (13)
The equation of the tangent t to the reflecting curve at the point M (x, y) is
Y − y = f ' ( x)( X − x) (14)
From equation (13)
y−a = R2 − x2 , (15)
8 A. Dvoretskii.
dy x
from here f ' ( x) = =− , (16)
dx 2
R − x2
and the tangent equation takes the form
x x2
Y =− X + y+ . (17)
R2 − x2 R2 − x2
Equation of the perpendicular dropped from the point A (x, 0) to the tangent t
R2 − x2
Y= ( X − x) . (18)
x
Fig. 6. Reflective curve is a circle and its orthotomics
Moreover, taking into account the direction of the incident rays perpendicular to the x-
axis, the x-coordinate of the point L is equal to the x-coordinate of the point M and the
y-coordinate of the point M is 0, since the straight line l coincides with the x-axis.
The point K of the intersection of the tangent t and the perpendicular to it is deter-
mined by the joint solution of equations (17) and (18)
x
XK = ( R2 − x2 y + R2 ) (19)
R2
y 2 (R 2 − x 2 )
YK = . (20)
R2
Figure 7 shows the result of the formation of four curves of the apparatus of reflec-
tions with a source at an infinity in which the reflecting curve is a circle.
The coordinates of the orthotomics point O (XoYo) are symmetric to the point L
with respect to the tangent t are
𝑋𝑂 = 2𝑋𝐾 − 𝑋𝐿 , (21)
Y0 = 2YK
(22)
Caustics, Orthotomics, and Reflecting Curve with Source at an Infinity 9
Fig. 7. The reflecting curve as a circle and its accompanying curves.
Substituting the coordinates of the points K and L into equations (21, 22), we obtain
the coordinates of the point of orthotomics
2 xy R 2 − x 2 + x R 2
XO = , (23)
R2
2 y( R 2 − x 2 )
YO = . (24)
R2
The x and y coordinates in equations (23) and (24) are the coordinates of a point on
the reflective curve m. In this example, the reflective curve is a circle.
Table 3. Determining of reflecting apparatus curves on the given reflecting curve as a circle
Reflecting Podera Orthothomic Caustic
curve
x p = R cos xo = − sin 2 xk = R cos 3
(
x = R cos 1 − (cos )2 − ) (b − R sin ) + y k = R sin
y = R sin − b sin cos + x + R cos
1
+ cos 2
y p = b (cos ) + y o = b cos 2 + 2
2
+ 2 R (sin )
3
+ R (sin )
3
10 A. Dvoretskii.
Conclusions
The studied reflective properties of plane curves as sections of reflecting surfaces are
preserved if these sections are generators of surfaces of revolution and rotative surfaces
(the normal to the section at a given point coincides with the normal to the surface at
the same point) and the source of the incident rays is in the same plane section.
The properties of the first reflection of curves can be investigated by their connec-
tions. It is giving us the possibility to construct new reflecting curves and reflecting
systems with the demand properties. Caustics, orthotomics, and reflective curves with
a source at infinity are a promising means of using the methods of geometric optics for
modeling natural objects in ergo design.
References
1. Cayley A. A memoir upon caustics// Phil. Trans. Roy. Soc. Lond. 147:273-312; Collected
Works. Vol. 2. pp. 336-380 (1857).
2. Glaeser G. Reflection on Spheres and Cylinders of Revolution. Journal for Geometry and
Graphics// Volume 3. No. 2, p. 121 – 139 (1999).
3. Bruce J.W., Giblin P.J. Curves and Singularities: A Geometrical Introduction to Singular-
ity Theory. Cambridge University Press. 262 p. (2012).
4. Bruce J.W., Giblin P.J., Gibson C.G. On caustics by reflection, Topology. I.21. 179 – 199
(1981).
5. Podgorny A.L., Dvoretsky A.T. Model of a plane reflective system for parallel incident rays//
Proceedings of the Tavriya State Agrotechnical Academy. – Issue 4, volume 15. –Melitopol : TSAA,
p.p. 12-15 (2002).
6. Dvoretsky A.T., Denisova T.V. Quasifocal lines// DOI: 10.30987/graphicon-2019-1-181-
184.