Surfaces and also Contour Plots

part 2: Quadric Surfaces

Quadric surfaces room the graphs the quadratic equations in three Cartesian variables in space. Favor the graphs of quadratics in the plane, their shapes depend on the signs of the assorted coefficients in your quadratic equations.

Spheres and also Ellipsoids

A sphere is the graph of an equation the the form x2+y2+z2=p2 for some real number p. The radius that the round is ns (see the number below). Ellipsoids space the graphs the equations the the form ax2+by2+cz2=p2, wherein a, b, and care all positive. In particular, a round is a an extremely special ellipsoid because that which a, b, and care all equal.

You are watching: Z=x^2+y^2 graph

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Plot the graph the x2+y2+z2=4 in your worksheet in Cartesian coordinates. Then choose different coefficients in the equation, and plot a non-spherical ellipsoid.

What curves execute you discover when you crossing a ball with a airplane perpendicular to among the coordinate axes? What perform you find for an ellipsoid?Paraboloids

Surfaces who intersections v planes perpendicular to any two the the name: coordinates axes space parabolas in those plane are called paraboloids. An instance is shown in the figure listed below -- this is the graph that z=x2+y2.

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make your very own plot of this surface ar in your worksheet, and rotate the plot to see it from miscellaneous perspectives. Monitor the suggestions in the worksheet. What are the intersections of the surface with planes of the form z=c, for some continuous c?

present that the intersections that this surface with planes perpendicular come the x- and y-axes are parabolas. change the equation come z=3x2+y2, and plot again. How does the surface ar change? In particular, what wake up to the curve of intersection v horizontal planes. The surface ar in the following number is the graph the z=x2-y2. In this case, the intersections v planes perpendicular to the x- and y-axes are still parabolas, but the 2 sets that parabolas differ in the direction in which lock point. For reasons we will see, this surface is referred to as a hyperbolic paraboloid -- and, for obvious reasons, that is also called a "saddle surface."

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make your very own plot the this hyperbolic paraboloid in your worksheet, and rotate the plot to see it from various perspectives. Monitor the proposal in the worksheet. What room the intersections the the surface with plane of the form z=c, because that some continuous c? define both parts of the name.

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Hyperboloids

Hyperboloids room the surface in three-dimensional room analogous come hyperbolas in the plane. Their specifying characteristic is that their intersections v planes perpendicular to any type of two the the name: coordinates axes space hyperbolas. There room two varieties of hyperboloids -- the first type is depicted by the graph of x2+y2-z2=1, i beg your pardon is presented in the number below. Together the number at the appropriate illustrates, this shape is very similar to the one generally used for nuclear power plant cooling towers. (Source: EPA"s response to 3 Mile Island Incident.)

This surface ar is referred to as a hyperboloid that one sheet since it is all "connected" in one piece. (We will gain to the other case presently.)

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do your very own plot that this surface in your worksheet, and also rotate the plot to view it from miscellaneous perspectives. Follow the suggestions in the worksheet. What are the intersections the the surface with airplane of the kind z=c, for some continuous c?

present that the intersections that this surface ar with plane perpendicular come the x- and also y-axes room hyperbolas. The other type is the hyperboloid of two sheets, and also it is shown by the graph the x2-y2-z2=1, displayed below.

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make your very own plot of this surface ar in her worksheet, and also rotate the plot to see it from various perspectives. Follow the suggestions in the worksheet. What room the intersections the the surface with planes of the kind z=c, because that some consistent c?

show that the intersections the these 2 surfaces with the ideal coordinate planes are hyperbolas.In every of these examples, the intersections that the surface ar with a household of planes tells united state a great deal about the structure of the surface. Us will return to this template in part 6, when we look at at contour lines.

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