Sec1Sess17

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Parametric curves and equations
More generally, parametric equations for arbitrary motion in the plane or in space.

Example: t (time) can be the parameter and functions for x, y, and z are parametric equations

Position vector: r(t) = x(t)i + y(t)j + z(t)k = <x(t),y(t),z(t)>. The vector from the origin to the moving point.

Distance vector: <math>OP = r(t) = x(t)i + y(t)j = <x,y></math>

Circle example:
Parametric form: <math>x(t) = a*cos(t) , y(t) = a*sin(t)</math>
Symmetric form: <math>x^2 + y^2 = a^2</math>

Eclipse example:
Parametric form: <math>x(t) = a*cos(t) , y(t) = a*sin(t)</math>
Symmetric form: <math>x^2/a^2 + y^2/a^2 = 1</math>

Line Example:
Parameterize the line through <math>(x_0,y_0)\mathrm{\ parallel\ to\ }\langle b_1,b_2 \rangle</math> by:
<math>\mathrm{x(t) = x_0 +t(b_1), y(t) = y_0 +t*b_2\ equivalent\ to\ r(t) = (x, y) = (x_0 +t(b_1), y_0 +t(b_2)) = (x_0, y_0)+t(b_1, b_2).}</math>

<math>\mathrm{Advanced\ line:\ parametric:\ x\ =\ 3(cos^2*t),\ y\ =\ 3(sin^2*t)\ .\ symmetric:\ x\ +\ y\ =\ 3,\ with\ x,\ y\ non-negative.}</math> (symmetric form looses info because the
line reverses back in direction on itself and that is not shown in that form.)

Helix: r(t) = cos(t)*i + sin(t)*j + a*t*k

Cycloid: wheel of radius a rolling on floor = x-axis

P = a point on the rim of a wheel starts at 0. What happens?
Curve with no simple symmetric form, but a practical parametric form is created

Question: position (x(t), y(t)) of the point P?

instead of t, theta is used. The (x(theta), y(theta)) is a function of the angle, theta, by which the wheel (circle) has rotated.

A = point where circle touches x-axis
a = radius of circle
B = circle center
O = origin (can be <0,0>)
T = <math>\mathrm{\theta}</math> , theta, angle of rotation
P = a point on the rim of a wheel starts at 0.
<math>\vec{OP}\ =\ \vec{OA}\ +\ \vec{AB}\ +\ \vec{BP}</math>

<math>\vec{OA} = \langle aT,0 \rangle </math>
this is the distance the circle has traveled on the x-axis
= arclength from A to P
the arclength along the circle once flattened equals the distance traveled by the circle on the x-axis

<math>\vec{AB} = \langle 0,a \rangle </math>
this is just a line that points strait up from the point the circle touches the x-axis

<math>\vec{BP}</math>, things known include |<math>\vec{BP}</math>| = a, angle T with vertical
= <-a*sin(T),-a*cos(T)>

<math>\vec{OP}</math> = <aT - a*sin(T), a - a*cos(T)>

* Finding the parametric equations for a point P on the circumfrance of a disk rotating over time t (cycloid):

1. start at origin
2. add <0,i> to go to center of disk where i is y at y-intercept
3. add <t,0> to go to center of disk at time t
4. add -<sin(angle),cos(angle)> to shift over by one unit at the given angle if movement is clockwise, adjust sin, cos to fit
if movement is counterclockwise
Finding angle: after time t the center of the disk has moved t units to the right, distance travelled
is r*angle where r is radius and angle is the central angle in radians. Therefore t = r*angle and angle = t/r.

OP(t) = <0, i> + <t, 0> + <− sin(t/r), − cos(t/r)>
since <0,i> applies to y axis than it is added on the y axis
since <t,0> applies to x axis than it is added on the x axis
r = r(t) = <t - sin(t/r), i - cos(t/r)>