Problem 44 - Calculus with Concepts in Calculus

Question

Sketch the quadric surface. \(z^{2}=x^{2}+4 y^{2}\)

Step-by-Step Solution

Answer

The surface is a cone with elliptical cross sections, symmetrical about the xy-plane, extending along the z-axis.

Step 1: Identify the Type of Quadric Surface

The given equation is \(z^2 = x^2 + 4y^2\). This equation is a form of an ellipsoid because it can be rewritten to resemble the general form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}\), but since the coefficients of \(x\) and \(y\) are not equal, and there is no \(z\) term added, it's specifically a cone.

Step 2: Rewrite the Equation in Standard Form

To understand the shape better, rewrite the equation as \(\frac{z^2}{1} = \frac{x^2}{1} + \frac{y^2}{\frac{1}{4}}\). This is the standard form of a cone, where the cross sections are ellipses.

Step 3: Analyze Cross Sections

Let's consider various cross sections of the surface:- For different constant values of \(z\), the cross-sections \(x^2+4y^2=z^2\) are ellipses.- For \(z=0\), the ellipse becomes a degenerate point at the origin.- As \(z\) increases positively or negatively, the size of the ellipses increases.

Step 4: Determine Symmetries and Axes

The surface is symmetric with respect to the \(xy\)-plane. The principal axis of the cone is along the \(z\)-axis due to the form \(z^2 = x^2 + 4y^2\). The surface opens in the positive and negative \(z\)-directions.

Step 5: Sketch the Surface

Draw the surface:1. Begin with the \(z=0\) plane marking the origin.2. Illustrate an increasing size of elliptical cross sections in both directions along the z-axis.3. The orientation is such that circles are widened in the y-direction due to the factor of 4 in \(4y^2\). The semi-minor axis is along the x-direction and the semi-major axis is along the y-direction.

Key Concepts

Conic Sections

Conic sections are curves obtained from the intersection of a cone and a plane. These curves include circles, ellipses, parabolas, and hyperbolas. Each of these curves has its own unique shape and properties. In the context of our exercise, the cross-sections taken parallel to the base of an elliptic cone represent conic sections. For example:

• If the intersecting plane is perpendicular to the cone's axis, the cross-section is a circle or an ellipse.
• When the plane intersects the cone parallel to one of its sides, a parabola is formed.
• If the plane cuts through both pieces of a double cone, a hyperbola appears.

Understanding conic sections is crucial to grasping the characteristics of more complex 3D objects, like the elliptic cone in the given exercise. The beauty of conic sections lies in their diverse applicability in physics, engineering, and architectural designs.

Elliptic Cone

An elliptic cone is a three-dimensional surface that extends infinitely in both directions along its principal axis. It is generated by revolving an ellipse around one of its axes. In simple terms, think of an elliptic cone like a party hat, but instead of a circular base, it has an elliptical one.
In the given exercise, the equation \[ z^2 = x^2 + 4y^2 \]represents an elliptic cone. The presence of different coefficients for \( x^2 \) and \( y^2 \) in the equation indicates the cone has an elliptical cross-section. This means that when you "slice" the cone at various planes parallel to its base, the shapes you see are ellipses.
The elliptic cone is symmetric around the principal axis, which, in our exercise, is along the \( z \)-axis. This symmetry indicates that any transformation, such as rotation around this axis, preserves the shape of the cross-sections, showing identical ellipses at equal distances above and below the base plane.

3D Geometry

Three-dimensional geometry is the study of shapes and objects that have height, width, and depth. It includes understanding how shapes like cubes, spheres, cylinders, and cones exist and relate to each other in space. In our context, the elliptic cone is a prime example of 3D geometry, showcasing how two-dimensional figures (like ellipses) create complex forms when extended along a dimension.
3D geometry helps in visualizing real-world objects and is crucial in fields like architecture, engineering, and computer graphics. When examining the elliptic cone described by \[ z^2 = x^2 + 4y^2 \],in the context of 3D space, note how this surface maintains a consistent orientation and symmetry as part of its geometric properties.

• The principal axis (\( z \)-axis in this case) is one key aspect that defines the orientation of this shape in space.
• The symmetry about this axis simplifies many calculations involving volumes, surface areas, and other geometric properties.

Recognizing how shapes like cones fit into 3D geometry enhances our spatial reasoning and ability to interact with and design in our multi-dimensional world.

Cross Sections

Cross sections are the 2D shapes we obtain when cutting through a 3D object with a plane. They provide insight into the structure and characteristics of the surface being studied. For the elliptic cone in our problem, cross sections play a critical role in understanding its geometry.

• When the plane is parallel to the base of our cone, the cross-sections become ellipses parallel to the \( xy \)-plane.
• At \( z = 0 \), the cross-section is simply a point at the origin, reflecting the cone's tip.
• As \( z \) moves away from zero, the size of the ellipses increases, providing a clearer picture of how the cone "opens" outward.

Examining cross sections helps us to see how the elliptic cone varies at different points along the \( z \)-axis. They are invaluable not only in mathematics but also in practical fields like medicine (CT scans rely on analyzing cross sections) and civil engineering, where they aid in structural analysis.