# Math 2060 - Section 12.6

## 1999 days ago by jimlb

Section 12.6

We give some of the basic graphs of cylinders and quadric surfaces.  If you would like to put the surfaces into 3-d so that you can use your glasses, click to make the surface interactive, right click on the surface and go to style -- > stereographic -- > red + cyan glasses.

var('x,y,t');
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(x, y, t\right)$

Example:  Here we graph the elliptic paraboloid $z = x^2 + 4 y^2$.  Note that the red curve is the ellipse $1/2 = x^2 + 4 y^2$ which occurs on the surface where $z = 1/2$.  The green curve is the parabola $z= 4y^2$ which occurs on the surface where $x = 0$.

P=plot3d(x^2 + 4*y^2, (x,-1,1), (y,-.5,.5), aspect_ratio=true); Q=parametric_plot3d(((1/sqrt(2))*cos(t), (1/sqrt(8))*sin(t), 0.5), (t, 0, 2*pi),color='red', thickness=5);R=parametric_plot3d((0, t, 4*t^2), (t, -.5, .5),color='green', thickness=5);(P+Q+R).show()

Example: Here we graph the ellipsoid $4 x^2 + y^2 + 4 z^2 = 4$.  The ellipse $y^2 + 4z^2 = 3$ is given in red.  This occurs on the surface when $x = 1/2$. The green curve is the circle $x^2 + z^2 = 1$, which occurs on the surface when $y = 0$.

var('s,t')
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(s, t\right)$
P=parametric_plot3d([sin(s)*sin(t), 2*cos(s)*sin(t), cos(t)], (s,0, 2*pi), (t, 0, 2*pi),plot_points=[50,50],aspect_ratio=true); Q = parametric_plot3d([0.5, sqrt(3)*cos(t), (sqrt(3)/2)*sin(t)], (t,0,2*pi), color='red', thickness=5);R = parametric_plot3d([cos(t), 0, sin(t)], (t,0,2*pi), color='green', thickness=5);(P+Q+R).show()

Example:  In this example we graph the cone $x^2 = y^2 + 6 z^2$. Note since we use a parametric plot for this, it appears there is nothing at the origin when there should be.  The red curve is the ellipse $9 = y^2 + 6z^2$, which occurs when $x =3$ on the cone.  The green curve is the hyperbola $x^2 - 6 z^2 = 1$, which occurs when $y = 1$.

P=parametric_plot3d([sqrt(s^2 + 6*t^2), s, t], (s,-4, 4), (t, -2, 2),plot_points=[50,50], aspect_ratio=true);Q= parametric_plot3d([-sqrt(s^2 + 6*t^2), s, t], (s,-4, 4), (t, -2, 2),plot_points=[50,50], aspect_ratio=true); R=parametric_plot3d([3, 3*cos(t), (3/sqrt(6))*sin(t)], (t, 0, 2*pi), color='red', thickness=5); S1=parametric_plot3d([sqrt(1+6*t^2), 1, t], (t, -2, 2), color='green', thickness=5);S2=parametric_plot3d([-sqrt(1+6*t^2), 1, t], (t, -2, 2), color='green', thickness=5);(P+Q+R+S1+S2).show()

Example: This example graphs the hyperboloid of two sheets given by $-x^2 + 4y^2 - z^2 = 4$. The red curve is the circle $36 = x^2 + z^2$, which occurs on the surface when $y = \sqrt{10}$. The green curve is the hyperbola $4y^2 - z^2 = 4$, which occurs on the surface when $x = 0$.

P1=parametric_plot3d([s, sqrt((s/2)^2 + (t/2)^2 + 1), t], (s,-20,20), (t, -20, 20)); P2=parametric_plot3d([s,- sqrt((s/2)^2 + (t/2)^2 + 1), t], (s,-20,20), (t, -20, 20));Q= parametric_plot3d([6*cos(t), sqrt(10), 6*sin(t)], (t,0,2*pi), color='red', thickness=5);R1 = parametric_plot3d([0, sqrt((t/2)^2+1), t], (t,-20,20), color='green', thickness=5);R2 = parametric_plot3d([0, -sqrt((t/2)^2+1), t], (t,-20,20), color='green', thickness=5);(P1+P2+Q+R1+R2).show()

Example:  The last example is the hyperboloid $y = z^2 - x^2$.  The red curve is the parabola $y = -x^2$, which occurs on the surface when $z = 0$.  The green curve is the parabola $y = z^2$, which occurs on the surface when $x =0$.  The yellow curve occurs when $y = 0$ and is the pair of lines $z = x$ and $z = -x$.

P=parametric_plot3d([s,t^2 - s^2, t], (s,-5,5), (t,-5,5));Q=parametric_plot3d([t,-t^2, 0], (t,-5,5), color='red', thickness=5);R=parametric_plot3d([0,t^2, t], (t,-5,5), color='green', thickness=5);S1=parametric_plot3d([t,0, t], (t,-5,5), color='yellow', thickness=5);S2=parametric_plot3d([t,0, -t], (t,-5,5), color='yellow', thickness=5);(P+Q+R+S1+S2).show()