2.6 Quadric surfaces  (Page 6/15)

Consider the parabolic reflector described by equation $z=20x^2+20y^2.$ Find its focal point.

Show that quadric surface $x^2+y^2+z^2+2xy+2xz+2yz+x+y+z=0$ reduces to two parallel planes.

Show that quadric surface $x^2+y^2+z^2-2xy-2xz+2yz-1=0$ reduces to two parallel planes passing.

[T] The intersection between cylinder ${\left(x-1\right)}^2+y^2=1$ and sphere $x^2+y^2+z^2=4$ is called a Viviani curve .

1. Solve the system consisting of the equations of the surfaces to find the equation of the intersection curve. ( Hint: Find $x$ and $y$ in terms of $z.)$
2. Use a computer algebra system (CAS) to visualize the intersection curve on sphere $x^2+y^2+z^2=4.$

Hyperboloid of one sheet $25x^2+25y^2-z^2=25$ and elliptic cone $\mathrm{-25}{x}^2+75y^2+z^2=0$ are represented in the following figure along with their intersection curves. Identify the intersection curves and find their equations ( Hint: Find y from the system consisting of the equations of the surfaces.)

[T] Use a CAS to create the intersection between cylinder $9x^2+4y^2=18$ and ellipsoid $36x^2+16y^2+9z^2=144,$ and find the equations of the intersection curves.

[T] A spheroid is an ellipsoid with two equal semiaxes. For instance, the equation of a spheroid with the z -axis as its axis of symmetry is given by $\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{c^2}=1,$ where $a$ and $c$ are positive real numbers. The spheroid is called oblate if $c<a,$ and prolate for $c>a.$

1. The eye cornea is approximated as a prolate spheroid with an axis that is the eye, where $a=8.7\text{mm and}c=9.6\text{mm}.$ Write the equation of the spheroid that models the cornea and sketch the surface.
2. Give two examples of objects with prolate spheroid shapes.

[T] In cartography, Earth is approximated by an oblate spheroid rather than a sphere. The radii at the equator and poles are approximately $3963$ mi and $3950$ mi, respectively.

1. Write the equation in standard form of the ellipsoid that represents the shape of Earth. Assume the center of Earth is at the origin and that the trace formed by plane $z=0$ corresponds to the equator.
2. Sketch the graph.
3. Find the equation of the intersection curve of the surface with plane $z=1000$ that is parallel to the xy -plane. The intersection curve is called a parallel .
4. Find the equation of the intersection curve of the surface with plane $x+y=0$ that passes through the z -axis. The intersection curve is called a meridian .

[T] A set of buzzing stunt magnets (or “rattlesnake eggs”) includes two sparkling, polished, superstrong spheroid-shaped magnets well-known for children’s entertainment. Each magnet is $1.625$ in. long and $0.5$ in. wide at the middle. While tossing them into the air, they create a buzzing sound as they attract each other.

1. Write the equation of the prolate spheroid centered at the origin that describes the shape of one of the magnets.
2. Write the equations of the prolate spheroids that model the shape of the buzzing stunt magnets. Use a CAS to create the graphs.

[T] A heart-shaped surface is given by equation ${\left(x^2+\frac{9}{4}{y}^2+z^2-1\right)}^3-x^2{z}^3-\frac{9}{80}{y}^2{z}^3=0.$

1. Use a CAS to graph the surface that models this shape.
2. Determine and sketch the trace of the heart-shaped surface on the xz -plane.

[T] The ring torus symmetric about the z -axis is a special type of surface in topology and its equation is given by ${\left(x^2+y^2+z^2+R^2-r^2\right)}^2=4R^2\left(x^2+y^2\right),$ where $R>r>0.$ The numbers $R$ and $r$ are called are the major and minor radii, respectively, of the surface. The following figure shows a ring torus for which $R=2\text{and}r=1.$

1. Write the equation of the ring torus with $R=2\text{and}r=1,$ and use a CAS to graph the surface. Compare the graph with the figure given.
2. Determine the equation and sketch the trace of the ring torus from a. on the xy -plane.
3. Give two examples of objects with ring torus shapes.