Show that the curve with parametric equations x=sint y=cost, z=sin^2 t is the curve of intersect

Show that the curve with parametric equations x=sint y=cost,...

Key Concepts

Visualization Through Projection

Visualization techniques, such as projecting a space curve onto coordinate planes or along given surfaces, can simplify the process of sketching. By understanding how a curve lies on and interacts with familiar surfaces, such as cylinders or paraboloids, one can more readily infer key features of its shape, such as symmetry, bounds, and overall trajectory.

Intersection of Surfaces

When a curve is defined as the intersection of two surfaces, every point on the curve satisfies the equations of both surfaces simultaneously. This concept is useful because it allows one to leverage the properties of the intersecting surfaces to understand and sketch the behavior of the curve, as well as to analyze its shape and orientation in three-dimensional space.

Elimination of the Parameter

Eliminating the parameter from a set of parametric equations involves rewriting the equations so that they no longer explicitly contain the parameter. This process helps to reveal the underlying Cartesian or implicit form of the curve, making it easier to identify its relationship with other surfaces and to analyze its geometric properties.

Parametric Representation of Curves

A parametric representation expresses the coordinates of points on a curve as functions of one or more parameters. This method allows one to describe curves in space in a way that can simplify the process of deriving properties of the curve, such as its tangent vectors, curvature, and intersections with other geometric objects.

Transcript

00:01 We are given the parametric equations of the curve, and we are given two surfaces, and we are asked to show that the curve with these parametric equations is the curve of intersection of the two surfaces.

00:16 And then, to sketch the curve.

00:21 Parametric equations for the curve are x equals sine t, y equals cosine t, and z equals sine squared t, and the two surfaces are z equals x squared.

00:35 And x squared plus y squared equals 1.

00:44 First, let's find the curve of intersection of the two surfaces.

00:52 So we have this since z is equal to x squared, and x squared plus y squared is equal to 1, we can substitute to get z is equal to 1 minus y squared.

01:12 And this is a parabola if we project onto the yz plane.

01:38 And so this is actually a parabolic parabola, and it lies on cylinder x squared plus y squared equals 1.

02:15 So recall that this is a parabolic cylinder.

02:44 This is just a regular cylinder.

03:00 So it makes sense that their intersection is going to be parabola.

03:06 And so we have that show that this is the same curve.

03:24 Notice that we have that x squared plus y squared is equal to one, and we have that z is equal to sine squared t, which is the same as x squared.

03:44 So we have the same equations and therefore the same curve of intersection.

03:52 Now to sketch a graph of this curve, we have our x, y, and z axes.

04:12 And i'll draw both the parabolic cylinder and the cylinder first.

04:19 And then i'll indicate their curve of intersections.

04:23 So the parabolic cylinder are graph in red...

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