There are a few ways to determine if a parametric equation describes a curve that is at rest or stationary:
Check if the x and y coordinates are constant
The simplest way is to check if the x and y coordinates given by the parametric equations are constant values, independent of the parameter t. For example:
x = 3
y = 5
Since x and y do not change with t, this curve is just a single point at (3, 5), so it is at rest.
Take the derivative with respect to t
Take the derivative of the x and y equations with respect to the parameter t. If the derivatives dx/dt and dy/dt both equal 0 for all values of t, then the curve is not moving or changing – it is at rest.
For example, consider the equations:
x = 5t
y = 2t + 3
Taking the derivatives:
dx/dt = 5
dy/dt = 2
Since dx/dt and dy/dt are not 0, this curve is not at rest.
Set the velocity vector equal to 0
The velocity vector for a parametric curve is given by:
Velocity = <dx/dt, dy/dt>
If we set this velocity vector equal to the 0 vector <0, 0>, then we know the curve is at rest. For example:
x = 3cos(t)
y = 4sin(t)
Velocity = <dx/dt, dy/dt> = <-3sin(t), 4cos(t)>
Setting this equal to <0, 0> shows that the only solution is t = 0. Therefore, the curve is momentarily at rest at this single position.
Conclusion
In summary, the main methods to determine if a parametric curve is at rest are:
- Check if x and y are constant values
- Take the derivative dx/dt and dy/dt and see if they are 0
- Set the velocity vector <dx/dt, dy/dt> equal to 0 and solve
If any of these conditions are satisfied, the parametric equation describes a curve that is stationary or at rest.