Parametric Equations Flashcards
Parametric curve
A set of points generated by a parametric equation
Parameter
The third variable, usually ‘t’ upon which the independent or dependent variable depends
Forward orientaion
The direction in which the curve is generated as the parameter increases
Derivatives of Parametric curves
Written as [dy/dt]/[dx/dt]
h’(t)/g’(t)
h(t)- function that outputs the y-value
g(t)- function that outputs the x-value
Parametric equations for an ellipse/circle
x(θ)=cos(θ)*‘r’
y(θ)=sin(θ)*‘r’
Parametric equations for a hyperbola
x(θ)=sec(θ)‘r’
y(θ)=tan(θ)‘b’
b= √(c^2-a^2) a= vertices of (0,±a) or (±a,0) c= foci at (0,±a) or (±a,0)
Parametric equations for a parabola
x(θ)=2θ*r
y(θ)=θ^2*r
Parametric equations in colloidal dispersion/concentration
S
Translations and rotations of axis
A
Celestial orbit in Parametric equations
S
Length of the polar curve
L= ∫ √(f(θ)^2+f’(θ)^2)dθ
Arc Length with time
s(t)=∫|v(u)|du
from a to t
Curvature
How much the tangent vector changes given time
Curvature (given arc-length)
k=|d(r’/|r’|)/ds|
Curvature (given time)
k=|d(r’/|r’|)/dt|/|v|
Also
=|T’(t)/r’(t)|