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Elementary Curves

Reference data and engineering information about elementary curves for mathematics applications.

elementarycurvesData Table

Overview

Engineering reference data for Elementary Curves in mathematics.

Key Formulas

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Roots of ax² + bx + c = 0.

Pythagorean Theorem

c2=a2+b2c^2 = a^2 + b^2

Right triangle relationship.

Circle Area

A=πr2A = \pi r^2

Area of a circle.

Logarithm

log⁡b(x)=ln⁡(x)ln⁡(b)\log_b(x) = \frac{\ln(x)}{\ln(b)}

Change of base formula.

Variables

Symbol Description Unit
π\pi Pi 3.14159...
ee Euler's number 2.71828...

Degenerate Conics

Degenerate conics are limiting cases of the standard (non-degenerate) conic sections where the intersection plane passes through the apex of the cone. They result in straight lines or points.

Parallel Lines

The equation x2−a2=0x^2 - a^2 = 0 represents a pair of parallel lines.

  • Parameter: aa is half the distance between the two parallel lines.

Intersecting Lines

The equation x2a2−y2b2=0\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 represents a pair of intersecting lines.

  • Parameter: The angle between the lines is 2arctan⁡(ba)2\arctan\left(\frac{b}{a}\right).

Coincident Lines

The equation x2=0x^2 = 0 represents a single line (or a pair of coincident lines).

References