# 3D SURFACE PLOTTER

### Surfaces và Contour Plots

Part 2: Quadric SurfacesQuadric surfaces are the graphs of quadratic equations in three Cartesian variables in space. Lượt thích the graphs of quadratics in the plane, their shapes depend on the signs of the various coefficients in their quadratic equations.

Spheres và Ellipsoids

A sphere is the graph of an equation of the khung x2+y2+z2=p2 for some real number p The radius of the sphere is p. (see the figure below). Ellipsoids are the graphs of equations of the size ax2+by2+cz2=p2, where a, b, & care all positive. In particular, a sphere is a very special ellipsoid for which a, b, and care all equal.

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What curves vì chưng you find when you intersect a sphere with a plane perpendicular lớn one of the coordinate axes? What bởi vì you find for an ellipsoid?**Paraboloids**

Surfaces whose intersections with planes perpendicular khổng lồ any two of the coordinate axes are parabolas in those planes are called paraboloids. An example is shown in the figure below -- this is the graph of z=x2+y2.

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Show that the intersections of this surface with planes perpendicular to the x- and y-axes are parabolas.

**Hyperboloids**

Hyperboloids are the surfaces in three-dimensional space analogous to lớn hyperbolas in the plane. Their defining characteristic is that their intersections with planes perpendicular to any two of the coordinate axes are hyperbolas. There are two types of hyperboloids -- the first type is illustrated by the graph of x2+y2-z2=1, which is shown in the figure below. As the figure at the right illustrates, this shape is very similar khổng lồ the one commonly used for nuclear power nguồn plant cooling towers. (Source: EPA"s Response to Three Mile Island Incident.)

This surface is called a hyperboloid of one sheet because it is all "connected" in one piece. (We will get khổng lồ the other case presently.)

Make your own plot of this surface in your worksheet, and rotate the plot to see it from various perspectives. Follow the suggestions in the worksheet. What are the intersections of the surface with planes of the form z=c, for some constant c? Show that the intersections of this surface with planes perpendicular lớn the x- and y-axes are hyperbolas.

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Show that the intersections of these two surfaces with the appropriate coordinate planes are hyperbolas.In each of these examples, the intersections of the surface with a family of planes tells us a great giảm giá about the structure of the surface. We will return lớn this theme in Part 6, when we look at contour lines.

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