Elementary Surfaces

Overview

Engineering reference data for Elementary Surfaces in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Cylinder Surfaces

Cylinder surfaces are quadric surfaces where one variable is absent from the equation, resulting in a surface generated by moving a line parallel to that axis along a base curve.

Elliptic Cylinder

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

where $a, b$ = semi-axes of the elliptical cross-section

Circular Cylinder

$x^2 + y^2 = a^2$

where $a$ = radius of the circular cross-section

Hyperbolic Cylinder

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

where $a, b$ = parameters determining the hyperbolic cross-section

Parabolic Cylinder

$y^2 - 2lx = 0$

where $l$ = focal parameter

Degenerate Quadrics

Degenerate quadric surfaces occur when the equation factors into simpler components, representing planes or combinations of planes.

Parallel Planes

$x^2 - a^2 = 0$

where $a$ = half the distance between the parallel planes

Intersecting Planes

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0$

where $\arctan\left(\frac{b}{a}\right)$ = half the angle between the intersecting planes

Coincident Planes

$x^2 = 0$

This represents two planes that occupy the same position in space (i.e., a single plane with multiplicity two).

References

• Original Source