Conics and Quadrics - Examples and Constructions

Constructing conics and understanding their properties through specific examples reveals the interplay between algebra, geometry, and physics.

ExampleDrawing Ellipses: The String Method

An ellipse can be drawn using two pins (at the foci) and a string:

Fix pins at points $F_1$ and $F_2$
Tie a string of length $2a > |F_1F_2|$ to both pins
Pull the string taut with a pencil and trace the curve

This works because every point $P$ on the ellipse satisfies $|PF_1| + |PF_2| = 2a$ (constant sum of distances to foci).

Quadric surfaces in three dimensions include diverse shapes. The standard form after suitable rotation and translation:

\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \quad \text{(ellipsoid)}

Variations give hyperboloids of one and two sheets, elliptic and hyperbolic paraboloids, and cylinders. Each has distinct topological and geometric properties.

DefinitionRuled Surfaces

A ruled surface contains infinitely many straight lines. Remarkably, hyperboloids of one sheet and hyperbolic paraboloids are ruled—through each point pass two distinct lines lying entirely on the surface.

Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ (hyperboloid of one sheet)

This property makes them useful in architecture (e.g., cooling towers, saddle roofs).

ExampleConic Sections via Matrix Diagonalization

To classify the conic $x^2 + 4xy + y^2 + 2x - 3 = 0$ :

Write as $\mathbf{x}^T Q \mathbf{x} = 0$ with appropriate matrix $Q$
Diagonalize $Q$ via eigenvalues: if eigenvalues have same sign, it's an ellipse; opposite signs give hyperbola
Transform to principal axes to get standard form

This algorithmic approach unifies all conic classifications using linear algebra.

Confocal coordinate systems use families of confocal ellipses and hyperbolas as coordinate curves. In such systems, many PDEs (like Laplace's equation) separate variables, making them valuable in mathematical physics.

Kepler's laws of planetary motion fundamentally involve conics. His first law (planets move in elliptical orbits with the Sun at one focus) transformed astronomy. The second law (equal areas in equal times) relates to the conservation of angular momentum, while the third law (period squared proportional to semi-major axis cubed) reveals the gravitational force law.

Remark

Applications in Modern Technology:

GPS satellites: Use elliptical orbits calculated via conic geometry
Radio telescopes: Parabolic dishes focus signals at receivers
Lithography: Elliptical masks in semiconductor manufacturing
Medical devices: Lithotripsy uses ellipsoidal reflectors to break kidney stones

The quadric discriminant determines the type of quadric surface, analogous to the conic discriminant. The rank and signature of the associated matrix classify the surface into 17 distinct types (modulo affine transformations), revealing the rich taxonomy of second-degree surfaces.