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):

>	#________ LINES _________________________________________ line1 :=  spacecurve([ t,0,  t], t=-5..5, color = red): line2 :=  spacecurve([ t,0, -t], t=-5..5, color = black): plnl :=   implicitplot3d( y = 0 ,x=-5..5,y=-5..5, z=-5..5,                         style=patchnogrid, shading=ZGRAYSCALE ):

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#________ PARABOLA _________________________________________ par :=  spacecurve([ t, 1-(t^2)/4, 1+(t^2)/4 ],t=-4..4,                       color=blue, thickness=2): plnp :=  plot3d( 2- y, x=-5..5, y= -5..5,                       style=patchnogrid, shading=ZGRAYSCALE ): xap :=  spacecurve([ t, 1, 1  ],t=-5..5, color = COLOR(RGB, .3,.7, .3) ): yap :=  spacecurve([ 0, t, 2-t],t=-3..5, color = COLOR(RGB, .3,.7, .3) ):

>

#________ CIRCLES _________________________________________ cir :=  spacecurve([ 3*cos(t), 3*sin(t), 3 ],t=-4..4,                       color=red, thickness=2): plnc :=  plot3d( 3, x=-5..5, y= -5..5,                     style=patchnogrid,  color = COLOR(RGB,.67,.67,.75) ): xac :=  spacecurve([ t, 0, 3 ],t=-5..5, color = COLOR(RGB,.2,.2,.6) ): yac :=  spacecurve([ 0, t, 3 ],t=-5..5, color = COLOR(RGB,.2,.2,.6) ):

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pln1;

  1. Two Lines      Plane Intersects Cone Through Origin    

A double cone is a relatively simple three dimensional shape. However, there are several types of curves which come about as intersections of the cones and a single plane.

>	display([cone1, cone2], orientation = [25,75] );

This is the simplest intersection. If we cut the cones with a single plane which passes through the axes of symmetry of the cones, we get something like this.

>	display([cone1, cone2, line1,line2, plnl], orientation = [25,75] );

The intersection of these shapes is a pair of intersecting lines.

>	display([cone1, cone2, line1,line2], orientation = [25,75] );

>	display([cone1, cone2, line1,line2], orientation = [-50,40] );

>	display([line1,line2, plnl], orientation = [33,71] );

  2. Parabola       Plane Intersects Cone Parallel to Edge    

Now we intersect the plane with the cones in a different way. We are going to tilt the plane so that its "slope" is the same as the edge of the cone. (For example, we could keep the plane passing through the origin where the cones meet, and tilt it until it becomes tangent to the cones, then raise is straight up or down.) The resulting shape is a parabola. Notices that the plane only intersects one of the two cones in this case.

>	display([ par, plnp, cone1, cone2,xap,yap ], orientation = [25,75] );

>	display([ par, cone1, cone2 ], orientation = [25,75] );

>	display([ par, plnp,xap,yap ], orientation = [35,75] );

>	display([ par, plnp,xap,yap ], orientation = [ -90,135],axes = none );

  3. Parabola        Equation & Standard Graph                      

    Plain Vanilla Parabola

Here is a very simple parabola and some variations that occur when shifting and stretching it.

>	plot( x^2, x = -3..3, t4,bl);

>	display( plot( (x-3)^2 + 5, x = -3..6,  y = 0..20, t4,bl),          plot( x^2, x = -3..6, y = 0..20, t1,bl2, l2) );

>	display( plot( 3*x^2, x = -3..3,  y = 0..20, t4,bl),          plot( x^2, x = -3..3, y = 0..20, t1,bl2, l2) );

    Formal Definition

Here is the formal definition of a parabola... as the set of points equidistant between a point, (0,1) in this case, and a line, y = -1 in this case.

>

display( plot( (1/4)*x^2, x = -3..3,t4,bl),          plot( [[-3,-1],[3,-1]], t4, ro, l3),          pointplot( [0,1], ro, symbol = diamond, symbolsize = 30),          plot( [[-1,-1],[-1,1/4],[0,1]], t1, color = green, l2),          plot( [[-2.5,-1],[-2.5,25/16],[0,1]], t1, color = green, l2),          plot( [[ 1,-1],[1,1/4],[0,1]], t1, color = green, l2),          plot( [[ 2.5,-1],[ 2.5,25/16],[0,1]], t1, color = green, l2)  );

>

  4. Circle             Plane Intersects Cone Perpindicular to Axis of Cone    

Another way to intersect the cones is with a plane which is perpindicular to the axes of symmetry of the cones - much like chopping a carrot with a knife. The result is a circle.

>	display([ cir, plnc, cone1, cone2], orientation = [33,72] );

>	display([ cir, plnc, cone1, cone2,xac,yac ], orientation = [48,63] );

>	display([ cir, plnc, xac,yac ], orientation = [43,62] );

>	display([ cir, plnc, xac,yac ], orientation = [90,0] );

  5. Circle             Equation & Standard Graph  

Circles are very simple since one looks like another - except for the size and position.

>	implicitplot( x^2 + y^2 = 2^2, x = -3..3, y=-3..3,t4,bl,sc);

>	display( implicitplot(  x^2 + y^2 = 2^2,  x = -3..3, y=-3..3, t1,bl2, l2),          implicitplot( (x-2)^2 + (y-4)^2 = 2^2, x = -3..6,y=-3..7,                        t4,bl,sc));

>	display( implicitplot(  x^2 + y^2 = 2^2,  x = -3..3, y=-3..3, t1,bl2, l2),          implicitplot( (x+2)^2 + (y+3)^2 = 3^2, x = -6..6,y=-6..3,                        t4,bl,sc));

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