Problem 104 - Calculus Early Transcendentals

Question

Let \(E\) be the ellipsoid \(x^{2} / 9+y^{2} / 4+z^{2}=1, P\) be the plane \(z=A x+B y,\) and \(C\) be the intersection of \(E\) and \(P\). a. Is \(C\) an ellipse for all values of \(A\) and \(B\) ? Explain. b. Sketch and interpret the situation in which \(A=0\) and \(B \neq 0\). c. Find an equation of the projection of \(C\) on the \(x y\) -plane. d. Assume \(A=\frac{1}{6}\) and \(B=\frac{1}{2} .\) Find a parametric description of \(C\) as a curve in \(\mathbb{R}^{3}\). (Hint: Assume \(C\) is described by \(\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) and find \(a, b, c, d, e, \text { and } f .)\)

Step-by-Step Solution

Answer

**Short Answer:** The intersection curve between the given ellipsoid and plane will always be an ellipse, regardless of the values of A and B. If A = 0 and B ≠ 0, the intersection will be an ellipse parallel to the xz-plane. The equation of the projection of the intersection curve onto the xy-plane is ( $$\frac{x^2}{9} + \frac{y^2}{4} = 1 - (Ax+By)^2$$ ). The parametric description of the intersection curve for A = 1/6 and B = 1/2 is given by ( $$\langle at + b \sin t, c \cos t + d \sin t, e \cos t + f \sin t \rangle$$ ) after solving the system of equations obtained by substituting x, y, and z expressions into the ellipsoid and plane equations.

Step 1: Understanding the properties of an ellipsoid and its equation

An ellipsoid is a quadric surface in three-dimensional space defined by the following equation: ( $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$ ) in which a, b, and c are the lengths of the semi-axes along the x, y, and z-directions, respectively.

Step 2: Analyze the intersection of the ellipsoid and the plane

Given ellipsoid equation: ( $$\frac{x^2}{9} + \frac{y^2}{4} + z^2 = 1$$ ) and the plane equation: ( $$z = Ax + By$$ ), combine these equations to express the intersection curve as: ( $$\frac{x^2}{9} + \frac{y^2}{4} + (Ax+By)^2 = 1$$ )

Step 3: Determine if the intersection curve is an ellipse for all A and B

The intersection curve is an ellipse if the equation can be expressed in the following form: ( $$\frac{x^2}{M} + \frac{y^2}{N} + K(Ax + By) = 1$$ ) where M, N, and K are constants. Regardless of the values of A and B, the equation of the intersection curve is a quadratic equation in x and y, so the intersection curve will always be an ellipse. Therefore, the answer to question a is yes.

Step 4: Study the case when A=0 and B≠0

Let A = 0 and B ≠ 0, the plane equation becomes ( $$z = By$$ ). The intersection curve will be on the constant z plane. In this case, the intersection will be an ellipse parallel to the xz-plane. Sketching will show the ellipsoid and the plane at different heights for different values of B, all intersections being parallel to the xz-plane.

Step 5: Find the projection of the intersection onto the xy-plane

The projection of the intersection C onto the xy-plane can be found by eliminating z from the intersection equation: ( $$\frac{x^2}{9} + \frac{y^2}{4} + (Ax+By)^2 = 1$$ ) Set z = 0 (since the xy-plane has z = 0): ( $$\frac{x^2}{9} + \frac{y^2}{4} = 1 - (Ax+By)^2$$ ), which is the equation of the projection of the intersection curve.

Step 6: Parametric description of the intersection curve

Assume C has parametric description as ( $$\langle a \cos t + b \sin t, c \cos t + d \sin t, e \cos t + f \sin t \rangle$$ ), for A = 1/6 and B = 1/2, substitute x, y, and z expressions into the ellipsoid equation and plane equation Solve the resulting system of equations for a, b, c, d, e, and f. The resulting parametric description of the intersection curve will be: ( $$\langle at + b \sin t, c \cos t + d \sin t, e \cos t + f \sin t \rangle$$ ).

Key Concepts

Parametric Equations

Parametric equations are essential for describing geometric shapes and curves using parameters instead of relying on conventional algebraic equations. In the context of three-dimensional geometry, parametric equations allow us to express the coordinates \(x, y, z\) as functions of one or more parameters, often with respect to time \(t\). This technique gives flexibility in tracing the path of a point along a curve.
You can think of a parametric equation like a set of instructions guiding how to travel through the space. For instance, consider the representation \( \langle a \cos t + b \sin t, c \cos t + d \sin t, e \cos t + f \sin t \rangle \). Here, the functions of \( \cos t \) and \( \sin t \) control how the point moves along each axis. The parameters \(a, b, c, d, e,\) and \(f\) determine the shape and position of the curve. This allows us to model and visualize complex curves like the intersection of an ellipsoid and a plane.
This parametric representation is not only visually intuitive but also simplifies the computation and manipulation of curves in various applications, from computer graphics to pathfinding algorithms.

Projection onto Plane

Projection onto a plane is like creating a shadow of a 3D object on a flat surface. It's a method to reduce dimensionality, focusing on certain aspects of a shape. In the case of projecting an intersection curve from an ellipsoid onto the \(xy\)-plane, we're essentially removing the \(z\)-component and flattening the curve.
To achieve this, we modify the equation of the curve, essentially treating \(z\) as zero. Thus, the modified version of the intersection equation involves only the \(x\) and \(y\) variables. For example, starting with \( \frac{x^2}{9} + \frac{y^2}{4} + (Ax+By)^2 = 1 \,\) simplifying this equation by setting \(z = 0\) gives an expression \( \frac{x^2}{9} + \frac{y^2}{4} = 1 - (Ax+By)^2 \.\)
Understanding how this projection works is crucial in fields such as computer graphics, physics, and engineering, where 3D objects often need to be analyzed or presented in simplified views. By reducing complexity without losing essential features, projections help in comprehending and working with multi-dimensional data more effectively.

Ellipses in Three-dimensional Space

Ellipses in three-dimensional space exhibit fascinating properties and are often an important part of geometric modeling. Unlike the straightforward circles you find in 2D, ellipses in 3D can appear in various forms depending on their orientation and the intersecting planes involved. An ellipsoid, as defined, is essentially a stretched sphere which can be intersected by planes in many ways, often producing ellipses.
Given the ellipsoid \( \frac{x^2}{9} + \frac{y^2}{4} + z^2 = 1 \,\) if you cut this ellipsoid with any plane, say \( z = Ax + By \,\) the resulting intersection is generally an ellipse. This stems from the fact that the intersection of a plane with a quadratic surface usually results in conic sections, of which ellipses are a type.
The fundamental reason why any cross-section of an ellipsoid by a plane results in an ellipse lies in the mathematics of quadratic equations and in the symmetry of the ellipsoid itself. Understanding these sections and how to describe them parametrically opens doors to advanced analysis in engineering, physics, and computer modeling, providing a basis for interpreting phenomena ranging from celestial orbits to modern architecture design.