Plotting a dot product

Question

From a conic equation obtained from a dot product, I want to plot its 2D graph, while being able to manipulate the variable.

Manipulate[
   K = {Sin[t] Cos[p], Sin[t] Sin[p], Cos[t]};
   oP = {x, y, 1};
   noP = Assuming[Element[{x, y, z}, Reals], Simplify[Norm[oP]]];
   nK = Assuming[Element[{t, p, x, y, z}, Reals], Simplify[Norm[K]]];
   {(K.oP)^2 == (Cos[a]*nK*noP)^2},
   {a, Pi/16, Pi, Pi/16, ControlPlacement -> Left},
   {t, 0, \[Pi], \[Pi]/256, ControlPlacement -> Left},
   {p, 0, \[Pi], \[Pi]/256, ControlPlacement -> Left}
 ]

I do get the correct equation, but I am unable to plot it in any way when I take it separately.

ParametricPlot[ 1 == (1 + x^2 + y^2) Cos[\[Pi]/16]^2, {x, -14, 14}, {y, -14, 14}]

show the axes, but not the graph.

Only when I try with the == function do I get the graph, which is not really useful... What should I change?

Answer 1

Using lists you can show more stuff in Manipulate. You could use Column or Row for more pretty formatting.

Manipulate[K = {Sin[t] Cos[p], Sin[t] Sin[p], Cos[t]};
 oP = {x, y, 1};
 noP = Assuming[Element[{x, y, z}, Reals], Simplify[Norm[oP]]];
 nK = Assuming[Element[{t, p, x, y, z}, Reals], Simplify[Norm[K]]];

 Column[
   {
     {(K.oP)^2 == (Cos[a]*nK*noP)^2},
     ContourPlot[(K.oP)^2 == (Cos[a]*nK*noP)^2, {x, -14, 14}, {y, -14, 14}]
   }
  ],

 {a, Pi/16, Pi, Pi/16, ControlPlacement -> Left}, 
 {t, 0, \[Pi], \[Pi]/256, ControlPlacement -> Left}, 
 {p, 0, \[Pi], \[Pi]/256, ControlPlacement -> Left}
]