Parametric Curves

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A Parametric Curve is a function from an interval I mapping to Rn. We consider 2D plane curves and 3D space curves. As the parameter t varies in the interval I, the point with position vector x(t) traces out the curve.

The trace of a parametric curve is the image of the function x, and we can find this by evaluating x at some points in the domain. A curve that does not cross itself is called simple.

The Arc-Length map is the integral of |x(t)| from t0 to t.

The Velocity Vector or Tangent Vector is dx/dt = x'. If |x'| > 0 then the parametric curve is regular and we can define the unit tangent vector T = x' / |x'|.

Plane Curve - A plane curve is a parametric curve whose trace lies in R2.

Space Curve - A space curve is a parametric curve whose trace lies in R3.

Parametric curves can also be expressed using polar coordinates.