0. Code

>	restart; with(plots):

Warning, the name changecoords has been redefined

>	sc := 'scaling = constrained': t1 := 'thickness = 1': t4 := 'thickness = 4': l2 := 'linestyle = 2': l3 := 'linestyle = 3': bl := 'color = COLOR(RGB, .4,.5,.7)': bl2 := 'color = COLOR(RGB, .2,.25,.35)': ro  := 'color = COLOR(RGB, .9,.6,0)':

>	cone1 := cylinderplot( [r,theta,r],r = 0..5, theta=0..2*Pi,                    style = patchnogrid): cone2 := cylinderplot( [r,theta,-r],r = 0..5, theta=0..2*Pi,                    style = patchnogrid):

>

>

#________ ELLIPSES _________________________________________ plne :=  implicitplot3d( z = 2 - (1/2)*y, x=-5..5, y= -5..3,z=-5..5,                  style=patchnogrid,  shading=ZGRAYSCALE  ):                   ell := spacecurve( [ (4/sqrt(3))*cos(t),                      (8/3)*sin(t)-4/3,                                          2-(1/2)*((8/3)*sin(t)-4/3), t = 0..2*Pi],             color = COLOR(RGB, .9,.8,.2),thickness = 3 ): xae := spacecurve([ t, 0, 2 ],t=-5..5, color = COLOR(RGB,.2,.2,.6) ): yae := spacecurve([ 0, t, 2-(1/2)*t],t=-5..3,color=COLOR(RGB,.2,.2,.6)):

>	#________ HYPERBOLAS _________________________________________

>

plnh :=   implicitplot3d( y = 2 ,x=-5..5,y=-5..5, z=-5..5,                         style=patchnogrid, shading=ZGRAYSCALE ): hyp := spacecurve( [ t, 2, sqrt(4+t^2), t = -5..5],             color = COLOR(RGB, .9,.8,.2),thickness = 3 ): hyp2 := spacecurve( [ t, 2, -sqrt(4+t^2), t = -5..5],             color = COLOR(RGB, .9,.8,.2),thickness = 3 ): xah :=  spacecurve([ t, 2, 0 ],t=-5..5, color = COLOR(RGB,.2,.2,.6) ): yah :=  spacecurve([ 0, 2, t ],t=-5..5, color = COLOR(RGB,.2,.2,.6)):

>

  1. Ellipse           Plane Intersects Cone @ At Shallow Angle                           

If we cut a cone with a plane which is perpindicular to its axis of symmetry, we get a circle. However, if our cut is not quite perpindicular ... in fact, the "slope" of the plane is anything bigger than 0, and less than the slope of the cone's edge... we can not a circle, but an ellipse.

>	display([ ell, plne, cone1, cone2 ], orientation = [6, 90] );

>	display([ ell, plne, cone1, cone2 ], orientation = [46,72] );

>	display([ ell, plne, cone1, cone2,xae,yae ], orientation = [42,55] );

>	display([ ell, plne, xae,yae ], orientation = [57,59] );

  2. Ellipse            Equation & Standard Graph      

Here is a very simple ellipse and shifted variant.

>	implicitplot( (x/5)^2 + (y/2)^2 = 1, x = -5..5, y = -4..4, t4,bl,sc);

>	display(implicitplot( ((x-3)/5)^2+((y-3)/2)^2=1,x =-3..8,y=0..8,t4,bl),         implicitplot( (x/5)^2+(y/2)^2=1,x =-5..5,y=-4..4,t1,bl2,l2,sc) );

>	implicitplot( (x/4)^2 + (y/6)^2 = 1, x = -5..5, y = -6..6, t4,bl,sc);

    Formal Definition

An ellipse is defined in this way. We start with any two points, which are called focus points. The sum of the distances from a point to each of the two fixed focus points (or foci), is constant.

>

f := sqrt(5^2 - 2^2): display( implicitplot( (x/5)^2+(y/2)^2=1,x=-5..5,y=-4..4,t4,bl),          plot( [[-f,0],[f,0]], t4, ro, l3),          pointplot( {[f,0],[-f,0]}, ro,symbol=diamond,symbolsize=30),          plot( [[-f,0],[3,8/5],[f,0]], t1, color = green, l2),          plot( [[-f,0],[5/2,-sqrt(3)],[f,0]], t1, color = gray, l2),          plot( [[-f,0],[-4,6/5],[f,0]], t1, color = red, l2), sc);

>

f := sqrt(6^2 - 4^2): display( implicitplot( (x/4)^2+(y/6)^2=1,x=-4..4,y=-6..6,t4,bl),          plot( [[0,-f],[0,f]], t4, ro, l3),          pointplot( {[0,-f],[0,f]}, ro,symbol=diamond,symbolsize=30),          plot( [[0,-f],[2, sqrt(27)],[0,f]], t1, color = green, l2),          plot( [[0,-f],[-3/2,sqrt(55)*3/4],[0,f]], t1, color = gray, l2),          plot( [[0,-f],[1/2,-sqrt(7)*9/4],[0,f]], t1, color = yellow, l2),          plot( [[0,-f],[-7/2,-sqrt(15)*3/4],[0,f]], t1, color = red, l2), sc);

  3. Hyperbola     Plane Intersects Cone @ At Steep Angle                      

A hyperbola is created by having a plane intersect a cone where the plane is parallel to the axis of symmetry of the cone. We saw one example of this kind of thing already - when the plane passes through the axis of symmetry - we get two intersecting lines. If the plane is parallel to any plane passing through the axis of symmetry, we get a hyperbola, which is a pair of opposite but equal curves.

>	display([ plnh, cone1, cone2 ], orientation = [156,77] );

>	display([ hyp, hyp2, plnh, cone1, cone2 ], orientation = [142,75] );

>	display([ hyp, hyp2, plnh, cone1, cone2,xah,yah ], orientation = [142,75] );

>	display([ hyp, hyp2, plnh, cone1, cone2,xah,yah],orientation = [42,55]);

>	display([ hyp,hyp2, plnh, xah,yah ], orientation = [67,85] );

  4. Hyperbola      Equation & Standard Graph      

Here are some basic hyperbolas.

>	implicitplot( (x)^2 - (y)^2 = 1, x = -4..4, y = -4..4, t4,bl,sc);

>	implicitplot( (x/5)^2 - (y/3)^2 = 1, x = -10..10, y = -7..7, t4,bl,sc);

>	implicitplot( -(x/5)^2 + (y/3)^2 = 1, x = -10..10, y = -7..7, t4,bl,sc);

>

>

    Formal Definition

A hyperbola is defined in similarly to how an ellipse is defined - except it's the difference of the two distances rather than the sum which is constant.

>	f := sqrt(3^2 + 2^2): display( implicitplot( (x/3)^2-(y/2)^2=1,x=-6..6,y=-4..4,t4,bl),          plot( [[-f,0],[f,0]], t4, ro, l3),          pointplot( {[f,0],[-f,0]}, ro,symbol=diamond,symbolsize=30),          plot( [[-f,0],[5,8/3],[f,0]], t1, color = green, l2),          sc);

>	display( implicitplot( (x/3)^2-(y/2)^2=1,x=-6..6,y=-4..4,t4,bl),          plot( [[-f,0],[f,0]], t4, ro, l3),          pointplot( {[f,0],[-f,0]}, ro,symbol=diamond,symbolsize=30),          plot( [[-f,0],[-6,2*sqrt(3)],[f,0]], t1, color = red, l2),          sc);

  5. Ellipses & Hyperbolas on Same Graph  

If we draw the two hyperbolas on the same graph, which differ only in their signs along with their asymptotes, we get this interesting diagram.

>

display(implicitplot( (x/5)^2 - (y/3)^2 = 1,                        x = -10..10, y = -7..7, t4,bl,sc),          implicitplot( -(x/5)^2 + (y/3)^2 = 1,                       x = -10..10, y = -7..7, t4,bl,sc),           plot( (3/5)*x, x = -10..10, l3, color = green),           plot( -(3/5)*x, x = -10..10, l3, color = green), sc);

>

display(implicitplot( (x/5)^2 - (y/3)^2 = 1,                        x = -10..10, y = -7..7, t4,bl,sc),          implicitplot( -(x/5)^2 + (y/3)^2 = 1,                       x = -10..10, y = -7..7, t4,bl,sc),         implicitplot( (x/5)^2 + (y/3)^2 = 1,                       x = -10..10, y = -7..7, t4,ro,sc),          plot( [[-5,-3],[5,-3],[5,3],[-5,3],[-5,-3]], t1,                color=black, l3),          plot( (3/5)*x, x = -10..10, l3, color = green),          plot( -(3/5)*x, x = -10..10, l3, color = green), sc);