centro-surface of an Ellipsoid

From the 2023 Maple Conference Art Gallery

The picture demonstrates one of the most beautiful and mysterious surfaces in mathematics. First

studied and drawn by Arthur Cayley and then by many mathematicians after him. The presented picture

is obtained using Cartesian coordinates. Probably, it is the first time that cartesian coordinates are used

for this purpose. All the previous attempts as far as I know were done using curvilinear coordinates.

a := 4.5;
b := 3;
c := 1.5;
f := (x, y) -> c*sqrt(1 - x^2/a^2 - y^2/b^2);
K := (x, y) -> 1/(a^2*b^2*c^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^2);
H := (x, y) -> 1/2*abs(x^2 + y^2 + f(x, y)^2 - a^2 - b^2 - c^2)/(a^2*b^2*c^2*(x^2/a^4 + y^2/b^4 + f(x,
y)^2/c^4)^(3/2));
R_1 := (x, y) -> 1/(H(x, y) - (H(x, y)^2 - K(x, y))^(1/2));
R_2 := (x, y) -> 1/(H(x, y) + (H(x, y)^2 - K(x, y))^(1/2));
plot3d([[f(x, y), -f(x, y)], [a*cos(x), b*sin(x), 0], [a*cos(x), 0, c*sin(x)], [0, b*cos(x), c*sin(x)], [(-a^2 +
b^2)*cos(x)^3/a, (-a^2 + b^2)*sin(x)^3/b, 0], [(-a^2 + c^2)*cos(x)^3/a, 0, (-a^2 + c^2)*sin(x)^3/c], [0, (-
b^2 + c^2)*cos(x)^3/b, (-b^2 + c^2)*sin(x)^3/c], [0, (-a^2 + b^2)*cos(x)/b, (-a^2 + c^2)*sin(x)/c], [(a^2 -
b^2)*cos(x)/a, 0, (-b^2 + c^2)*sin(x)/c], [(a^2 - c^2)*cos(x)/a, (b^2 - c^2)*sin(x)/b, 0], [x - x*R_2(x,
y)/(a^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^(1/2)), y - y*R_2(x, y)/(b^2*(x^2/a^4 + y^2/b^4 + f(x,
y)^2/c^4)^(1/2)), f(x, y) - f(x, y)*R_2(x, y)/(c^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^(1/2))], [x - x*R_2(x,
y)/(a^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^(1/2)), y - y*R_2(x, y)/(b^2*(x^2/a^4 + y^2/b^4 + f(x,
y)^2/c^4)^(1/2)), -f(x, y) + f(x, y)*R_2(x, y)/(c^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^(1/2))], [x - x*R_1(x,
y)/(a^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^(1/2)), y - y*R_1(x, y)/(b^2*(x^2/a^4 + y^2/b^4 + f(x,
y)^2/c^4)^(1/2)), f(x, y) - f(x, y)*R_1(x, y)/(c^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^(1/2))], [x - x*R_1(x,
y)/(a^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^(1/2)), y - y*R_1(x, y)/(b^2*(x^2/a^4 + y^2/b^4 + f(x,
y)^2/c^4)^(1/2)), -f(x, y) + f(x, y)*R_1(x, y)/(c^2*(x^2/a^4 + y^2/b^4 + f(x, y)^2/c^4)^(1/2))]], x = -5 .. 5, y
= -5 .. 5, color = [5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 10, 10])

Maple Learn Document

The motivation of this Maple Art proposal is to popularize a less known chapter in the history of

mathematics: Cayley’s attempt to find the shape of the surface containing all the centers of the principal

curvatures of an ellipsoid. Cayley called it centro-surface of an ellipsoid. Nowadays, this surface is known

as focal surface of an ellipsoid, Cayley’s Astroida or by more popular name (because of applications in

Optics and related fields) caustic of an ellipsoid. Arthur Cayley’s paper from 1873 contains sophisticated

algebraic calculations followed by an attempt to draw this surface. He says: “I constructed on a large

scale a drawing of the centro-surface for the values 𝑎2 = 50, 𝑏2 = 25, 𝑐2 = 15. (These were chosen so that 𝑎, 𝑏, 𝑐 should have approximately the integer values 7, 5, 4, and that 𝑎2 + 𝑐2 should be well

greater than 2𝑏2 ; they give a good form of surface,…”

After A. Cayley there were many attempts to recreate this surface both as drawings and as 3-D models.

With the dawn of the computer graphics era the quality and accuracy of these recreations increased and

we can now find many interesting and colorful renderings of this surface in books, articles and websites.

More pictures and information about history and mathematics can be found in https://arxiv.org/abs/2305.06065