Shine a light on Conic sections

Author:
Will Le
Circle, ellipse, parabola, hyperbola and straight line (and point) are different shapes, somehow very different, but all belong to one family of conic sections (a.k.a. "conics"), i.e. the intersections between a cone and a plane. But why a cone? Why slicing a cone?
A light cone cast on the plane as a circle and an ellipse
A light cone cast on the plane as a circle and an ellipse
That's because of the projective nature of the light cone.
Image
◯ The cone and conics are all projections of circle the perfect shape:
  • A cone is the shape of a circle projected prom one point (light source) into 3D space, like when you point a flashlight to the sky.
  • A conic is the shape of a circle projected onto a plane (2D space), like when you shine a flashlight on a wall (to see ellipses) or on the street (to see hyperbolas). Because the boundary of the light cone of a normal (non-laser) flashlight is blur, it's better to place a coin in front of your light to see its shadow cast on wall or table top instead.
So, circle is the generator of all conics! Let's interact with this 3D applet to explore the conics: Click button to pause animation; Drag the point "Light direction" around; Adjust aperture angle; Turn on focal spheres of Dandelin, unit sphere; Jumps to parabola, etc.
A light cone cast on the plane as a parabola and a hyperbola
A light cone cast on the plane as a parabola and a hyperbola
With aperture angle in range 0° < αo < 180° and projection angle αp ranging from 0° (perpendicular to the plane) to 90° (parallel to the plane), we have following projected images.
  • Circle at αp = 0°: It's the original shape, only scaled (up).
  • Ellipse when 0° < αp < 90° - : It's the slant view of circle that we usually .
  • Parabola at αp + = 90°: It's the transitional shape between ellipse and hyperbola, when the upper edge of light cone is parallel to the plane. It still comprises a single component like the ellipse, but it's already been open like the hyperbola.
  • Hyperbola when αp + > 90°: Its two branches are projection of the two opposite cones.
Note: Here projection angle αp is defined relative to the normal vector of the plane, i.e. αp is the incidence angle of the center ray in light cone, or the angle of inclination in view cone.
Two degenerate cones cast on the plane as a straight line and a point (Fp)
Two degenerate cones cast on the plane as a straight line and a point (Fp)
With aperture angle αo at two extremes, we have two degenerate shapes:
  • Straight line at αo = 180°: It's a degenerate hyperbola, as the light cone degenerates into a plane.
  • Point at αo = 0°: It's a degenerate ellipse, as the light cone degenerates into a straight line.
The following image shows how the family of circles on a sphere cast on the plane as a family of conics as αo ranging from 0° to 180°. (A family of curves is called a "pencil".)
  • αo = 0°: The generator circle degenerates into a point (south pole) on the sphere, and casts to a point of degenerate ellipse on the plane (shown as red points).
  • 0° < αo < 180°: The generator circle is a small circle on the sphere, and casts to an ellipse, a parabola or a hyperbola on the plane. Smaller circles generate ellipses while larger circles generate hyperbolas, and the one in the middle generates parabola.
  • αo = 180°: The generator circle is a great circle (equator) on the sphere, and casts to a straight line of degenerate hyperbola on the plane (shown as read circle and red line).
Because every pair of antipodal points (two poles) on a sphere determines exactly one great circle (equator) and vice versa, the corresponding point PL and straight line (in red) on the projective plane make a dual of pole and polar (line). Duality (between point & line) and polarity (pole & polar) are very important feature of projective geometry.
A pencil of circles on the sphere casted on the plane as a pencil of conics
A pencil of circles on the sphere casted on the plane as a pencil of conics

View cone and the unified image of conics

Have you ever wondered why the slant view of a circle is an ellipse?
Crop circles in desert
Crop circles in desert
Here we have to deal with the reverse direction:
  • View cone (versus light cone) is the projection of a conic in 3D space back into one point (the eye).
  • For any conic, there's a view cone projecting it into a circle.
Although both light cone and view cone are circular (their bases are circle), different from light cone which is a right cone (its axis is perpendicular to the base), the view cone is in ingeneral an oblique cone (its axis make an angle with the base). This oblique cone is defined and used by Apollonius of Perga to unify the definition of conics. An oblique circular cone is also a right elliptic cone. So, at the back of our eye, because the main disk of retina is perpendicular to the eye's axis, we see the image of the view cone as an ellipse. Note: Don't confuse this perspective view cone with the viewing cone of an electronic screen.
Image
Let's look at this 3D applet to see how an arbitrary conic is projected into a circle (and an arbitrary quadrilateral is projected into a square).
Project an ellipse into a full circle
Project an ellipse into a full circle
Project a hyperbola into a circle (only the upper part of the circle is visible infront of the eye)
Project a hyperbola into a circle (only the upper part of the circle is visible infront of the eye)