How to plot a spiral on a plane?

Question

I have a problem where I am given a curve in two dimensions. I must plot this curve on a given plane with the starting point of the curve on a certain point on the plane. I have no problems plotting the plane but I do not understand how to move a curve and tilt it so it lies on the plane.

The curve is defined by {x[t], y[t]} = {t Cos[2 t], t Sin[2 t]}. The equation of the plane is 3 (x + 1) + 4 (y - 2) + (z - 2) = 0. The spiral must be centered at {-1, 2, 2}.

Answer 1

Since the given plane is already in a convenient form, we can easily extract the center point and the normal of the plane to use with RotationTransform[] and InfinitePlane[]. Thus,

With[{pt = {-1, 2, 2}, nrm = {3, 4, 1}}, 
     Show[ParametricPlot3D[pt + RotationTransform[{{0, 0, 1}, nrm}][
                           {t Cos[2 t], t Sin[2 t], 0}] // Evaluate, {t, 0, 2 π}], 
     Graphics3D[{{Red, AbsolutePointSize[8], Point[pt]},
                 {LightBlue,
                  InfinitePlane[pt, Rest[Orthogonalize[{nrm, {1, 0, 0}, {0, 1, 0}}]]]}}], 
  Lighting -> "Neutral"]]

Answer 2

The point:

p = {-1, 2, 2};
point = ListPointPlot3D[{p}, PlotStyle -> {Blue, PointSize[Large]}]

The plane:

eqPlane = 3 (x + 1) + 4 (y - 2) + (z - 2) == 0;

f[x_, y_] := z /. First @ Solve[eqPlane, z]

f[x, y]

7 - 3 x - 4 y

plane = ParametricPlot3D[{x, y, f[x, y]}, {x, -10, 10}, {y, -10, 10}, Mesh -> None, BoxRatios -> 1]

The spiral:

{a[t_], b[t_], c[t_]} := {t Cos[2 t], t Sin[2 t], f[a[t], b[t]]}

Origin of the spiral:

s0 = {a[0], b[0], c[0]}

{0, 0, 7}

Plot of the spiral moved to the point p:

spiral = ParametricPlot3D[{a[t], b[t], c[t]} - s0 + p, {t, 0, 2 π}, PlotStyle -> Red, BoxRatios -> 1]

Simplify[eqPlane]

3 x + 4 y + z == 7

view = Coefficient[Simplify[First@eqPlane], #] & /@ {x, y, z}

{3, 4, 1}

Result:

Show[spiral, plane, point, ViewPoint -> view]

Answer 3

An uninspiring way:

{x[t], y[t]} = {t Cos[2 t], t Sin[2 t]};
 p = ({x, y, z} /. First@Solve[(x + 1) + 4 (y - 2) + (z - 2) == 0, z])
q = p /. {x -> x[t], y -> y[t]};
Show[Plot3D[Evaluate[p[[3]]], {x, -10, 10}, {y, -10, 10}, 
  Mesh -> None, PlotStyle -> {LightBlue, Opacity[0.5]}], 
 ParametricPlot3D[Evaluate[q], {t, 0, 2 Pi}, PlotStyle -> Red]]