The surface of a paraboloid results from rotating a parabola about it's axis. The parametric form is s(u, v) = (v cos(u), v sin(u), bv^2). Explain how to derive the parametric form of the paraboloid from the equation of a parabola. Verify that the curvature of a paraboloid is k(u, v) = 4b/(4b^2v^4 +1).

The surface of a paraboloid results from rotating a parabola about it's axis. The parametric form is s(u, v) = (v cos(u), v sin(u), bv^2). Explain how to derive the parametric form of the paraboloid from the equation of a parabola. Verify that the curvature of a paraboloid is k(u, v) = 4b/(4b^2v^4 +1).

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.6: Parametric Equations

Problem 5ECP: Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle...

See similar textbooks

Similar questions

Find parametric equations for the line tangent to the helix
Find a parametric equation for the line that is perpendicular to the graph of the equation 4x^2 + 2y^2 +3z^2 = 39 at the points (1,2,3). Use t as the variable
Find parametric equations for a circle of radius 2, centered at (3, 5).
Find the parametric equation of this/
find the parametric equation (linear). remove the parameter to find the equation. a. x=6-3t^3 y=4+2t^3 b. z=6-3(root (t)), y=4+2(root(t))
Find parametric equations for the line formed by the intersection of planes x+y-z=3 and 3x-y+3z=5.
A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 650 km/hour along a line at 8500 meters above the line joining Denver and Bismark. Bismark is about 850 km in the direction 60° north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the z-axis points east and the y-axis points north, and that the earth is flat. Measure distances in kilometers and time in hours. F(t) =
A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 650 km/hour along a line at 8500 meters above the line joining Denver and Bismark. Bismark is about 850 km in the direction 60° north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the x-axis points east and the y-axis points north, and that the earth is flat. Measure distances in kilometers and time in hours. F(t) = 10.15 k + t
Find parametric equations and a parameter interval for the motion of a particle that starts at the point (-2, 0) in the xy-plane and traces the circle x2 + y2 = 4 three times clockwise.
Consider the following points: A(1, 1, −2), B(−1, 2, 1) and C(1, 0, 1). Determine theparametric equations of either the unknown line or the unknown plane in R^3 a. the line through the points A and B
A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 750 km/hour at a constant height of 7000 meters above the line joining Denver and Bismark. Bismark is about 850 km in the direction 60° north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the x-axis points east and the y-axis points north, and that the earth is flat. Measure distances in kilometers and time in hours. r(t) = ?
Find parametric equations for the line through the point (6,7,5) and parallel to the line: x-3/-2 = y+8/7 = z-4/5