Parametric Equations Flashcards

0
Q

Parametric curve

A

A set of points generated by a parametric equation

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1
Q

Parametric equation

A

An equation that either specifies x or y in terms of a third variable (usually ‘t’)

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2
Q

Parameter

A

The third variable, usually ‘t’ upon which the independent or dependent variable depends

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3
Q

Forward orientaion

A

The direction in which the curve is generated as the parameter increases

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4
Q

Derivatives of Parametric curves

A

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

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5
Q

Parametric equations for an ellipse/circle

A

x(θ)=cos(θ)*‘r’

y(θ)=sin(θ)*‘r’

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6
Q

Parametric equations for a hyperbola

A

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)
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7
Q

Parametric equations for a parabola

A

x(θ)=2θ*r

y(θ)=θ^2*r

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8
Q

Parametric equations in colloidal dispersion/concentration

A

S

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9
Q

Translations and rotations of axis

A

A

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10
Q

Celestial orbit in Parametric equations

A

S

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11
Q

Length of the polar curve

A

L= ∫ √(f(θ)^2+f’(θ)^2)dθ

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12
Q

Arc Length with time

A

s(t)=∫|v(u)|du

from a to t

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13
Q

Curvature

A

How much the tangent vector changes given time

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14
Q

Curvature (given arc-length)

A

k=|d(r’/|r’|)/ds|

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15
Q

Curvature (given time)

A

k=|d(r’/|r’|)/dt|/|v|
Also
=|T’(t)/r’(t)|

16
Q

Curvature (given a point)

A

k=(r”xr’)/(|v|^3)

17
Q

T-value of parametric equations

A

T=(r’/|r’|)