How to use
- Enter functions f (t),
g(t), and
h(t)
in the text input fields
marked "f (t)=",
"g(t)=", and
"h(t)=".
(Example:
f (t)=3cos t,
g(t)=3sin t,
h(t)=t/3)
- Click the "Graph" button
(this button also refreshes the graph).
- Rotate the graph by
clicking and dragging the mouse on the graph.
- Enter first derivative functions f '(t),
g'(t), and
h'(t)
in the text input fields
marked "f '(t)=", "g'(t)=",
and "h'(t)=", and click the "Graph"
button to refresh the graph.
(Example:
f '(t)=-3sin t,
g'(t)=3cos t,
h'(t)=1/3)
Examples
Helix:
f (t)=3cos t,
g(t)=3sin t,
h(t)=t/3, with
appropriate derivatives
Wavy circle:
f (t)=3cos(t),
g(t)=3sin(t),
h(t)=sin(2t), with appropriate derivatives
Other Notes
The parametric path itself is shown in blue.
The velocity vector
v(t)=<f '(t),
g'(t), h'(t)>
and acceleration vector
a(t)=<f ''(t),
g''(t), h''(t)>
are shown in orange.
The standard unit tangent vector T is defined to be the unit
vector in the direction of the velocity vector v(t).
The standard unit normal vector N is defined to be the unit
vector for which the acceleration vector a(t) lies
in the T-N plane, with component aN
(the length of the projection of a(t) onto N)
being positive.
The standard unit binormal vector B is defined so that
B=TxN.
The unit vectors T, N, and B are shown in
green.
Labels show the values of aT (the tangent component
of a(t)) and aN (the normal component
of a(t)).