Parametric Equations of Curves in Two Dimensions
Some of the common two dimensional curves with their parametric equation are given in the following table:
Curve | Normal Equation | Parametric Equation |
|---|---|---|
Line | ax + by = c | x =x0 +at and y = y0 + bt |
Circle | (x – h)2 + (y – k)2 = r2 | x = r cos t + h and y = r sin t + k |
Ellipse | (x – h)2/a2 + (y – k)2/b2 = 1 | x = a cos t + h and y = b sin t + k |
Parabola | Horizontal Parabola
Vertical Parabola
| Horizontal Parabola
Vertical Parabola
|
Hyperbola | (x – h)2/a2 – (y – k)2/b2 = 1 | x = a sec t + h and y = b tan t + k |
Cycloid | x = r arccos[(r-y)/r] – √(2ry – y2) | x = a(θ – sin θ) and y = a(1 – cos θ) |
Lissajous Curve | – | x = a cos (k1t) and y = b sin (k2t) |
Where,
- For the line, (x0, y0) is a point on the line, and a and b are the direction ratios.
- For the circle, (h, k) is the center of the circle and r is the radius.
- For the ellipse, (h, k) is the center of the ellipse, a is the length of the semi-major axis, b is the length of the semi-minor axis, and t is the parameter.
- For the parabola, (h, k) is the vertex of the parabola and a determines the direction of the opening.
- For the hyperbola, (h, k) is the center of the hyperbola, a is the distance from the center to a vertex along the x-axis, b is the distance from the center to a vertex along the y-axis, and t is the parameter.
Parametric Equations
Parametric equations are a way to describe curves and shapes using one or more parameters. Instead of expressing coordinates directly, we use these parameters to define how points move along the curve. This method offers flexibility in representing complex curves and analyzing their behaviour, making it useful in various fields like mathematics, physics, engineering, and computer graphics.
Table of Content
- Parametric Equations Definition
- Parametric Function Definition
- Parametric Curve Definition
- Properties of Parametric Equations
- Applications of Parametric Equations