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The purpose of this lab is to acquaint you with some common three-dimensional shapes.


Three-dimensional curves can be entered as a function (or functions) of two variables or as an expression.
Remember the definition of a function when entering your shape. For example, the sphere can be entered as two functions or as one implicit expression.
To look at the cross-section of the sphere you cut the sphere along a plane - i.e. you hold a variable constant. So the intersection of the sphere and the $z=\frac{1}{2}$ plane is:
> implicitplot(subs(z=1/2,h),x=-1..1,y=-1..1);
Notice that the plot is a two-dimensional circle. To intersect vertical planes hold the $x$ or $y$ constant.
> implicitplot({subs(x=1,h),subs(x=1/3,h)},y=-1..1,z=-1..1,labels=[y,z]);
Other three-dimensional shapes can be made from known conic sections. A few of these will be analyzed in the exercises.


(Note: In all plots include the option scaling=constrained).
  1. For the given equations below, plot two dimensional level curves parallel to the $xy$ plane and then plot two dimensional cross sections in the $xz$ plane and the $yz$ plane. Identify the type or shape of the quadric surface, ie. a sphere, cylinder, cone, elliptic cone, paraboloid, elliptic parabaloid, ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, or a hyperbolic parabaloid (saddle). Once you have determined the shape of the surface, supply a three dimensional plot to support your conclusion.

    \begin{displaymath}z=x^2+y^2 \end{displaymath}


    \begin{displaymath}z=x^2-y^2/4 \end{displaymath}


    \begin{displaymath}x^2+y^2-z^2=4 \end{displaymath}


    \begin{displaymath}x^2-y^2-z^2=4 \end{displaymath}

next up previous
Next: About this document ... Up: lab_template Previous: lab_template
Jane E Bouchard