Multivariable Calculus: Parameterize the curve of intersection
In this exercise, we parameterize the curve of intersection between the plane z=2x+2 and the paraboloid z=x^2+y^2-1. The curve is a tilted circle. By combining the equations of the two surfaces, we find a parametric description for the curve: r(t) = ( 1 + 2cos(t) , 2sin(t) , 4 + 4cos(t) ). The parameter t ranges from 0 to 2pi, allowing us to trace the entire curve of intersection.
The approach involves substituting the plane equation into the paraboloid equation to eliminate the z-coordinate. Completing the square shows that the curve lies on a circle with radius 2, centered at (1, 0). Using polar coordinates, we parametrize the x and y coordinates using trigonometry. Then the plane gives us the appropriate equation for the z-coordinate.
I conclude by mentioning a demonstration that will show the surfaces intersecting and the curve being traced out (with MATLAB).
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