Equations of a Curve Space: An Ellipse

These are pictures/animations to visualize the vector and parametric of a curve. The curve is an ellipse. The first two animations may take a while to load.

The first picture represents the vector equation r(t)=<cos(t), sin(t), 2cos(t)>.The animation is from t=-2π to t=2π.

The second picture represents the parametric equations x=cos(t), y=sin(t), z=t. The animation is again from t=-2π to t=2π.

The ellipse sits on three surfaces as shown below. The arbitrary point (cos(t), sin(t), 2cos(t)) on the ellipse satisfies all three surface equations regardless of the value of t. So all points of the spiral are on each of those surfaces. You can get the equations of the surfaces by trying to eliminate the t in the equations.

A curve in space can also be described as the intersection of two surfaces. Below are the three possibilities: