Consider the paraboloid z=x^2+y^2. The plane 3x-2y+z-7=0 cuts the paraboloid, its intersection being a curve.
What is the "the natural" parametrization of this curve?
Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your starting point, give the parametrization of the curve on the surface.
To find the natural parametrization of the curve formed by the intersection of the plane and paraboloid, we first need to parametrize the circle in the xy-plane.
The equation of the plane is 3x - 2y + z - 7 = 0. To find the equation of the circle in the xy-plane, we set z = 0.
Now, substitute z = 0 into the equation of the plane: 3x - 2y + 0 - 7 = 0. Simplifying this equation gives us: 3x - 2y = 7.
This equation represents a circle in the xy-plane. To parametrize the circle, we can use the equations:
x = r * cos(t)
y = r * sin(t)
where r is the radius of the circle and t is the parameter that ranges from 0 to 2*pi, traversing the circle counterclockwise exactly once.
From the equation 3x - 2y = 7, we can see that the coefficients of x and y are 3 and -2 respectively. This means the circle intersects the x-axis at x = 7/3, which is the point with the largest x coordinate.
Therefore, the radius of the circle is the distance from the origin (0, 0) to the point (7/3, 0). Using the distance formula, we have:
r = sqrt((7/3)^2 + 0^2) = 7/3.
Now we can write the parametric equations for the circle in the xy-plane:
x = (7/3) * cos(t)
y = (7/3) * sin(t)
To find the parametrization of the curve on the paraboloid surface, we substitute these values of x and y into the equation of the paraboloid: z = x^2 + y^2.
z = (7/3)^2 * cos^2(t) + (7/3)^2 * sin^2(t)
z = (49/9) * (cos^2(t) + sin^2(t))
z = (49/9)
Therefore, the natural parametrization of the curve on the paraboloid surface is:
x = (7/3) * cos(t)
y = (7/3) * sin(t)
z = (49/9), where t ranges from 0 to 2*pi.