Image of a point in a plane: For point P(x0,y0,z0) and plane ax+by+cz=d:

1. The perpendicular from P to the plane: $\frac{x-x0}{a}$ = $\frac{y-y0}{b}$ = $\frac{z-z0}{c}$ = t.
2. General point on this line: (x0+at, y0+bt, z0+ct).
3. Foot of perpendicular: substitute in plane equation to find t = $\frac{d-ax0-by0-cz0}{(a^2+b^2+c^2)}$.
4. Foot F = (x0+at, y0+bt, z0+ct) with the computed t.
5. Image P' = 2F - P.

Image of a point in a line: For point P and line through A with direction b:

1. Foot F = A + tb where t = (AP.b)/|b|^2.
2. Image P' = 2F - P.

Reflection of a line in a plane: Find the image of any point on the line (not on the plane) and the intersection of the line with the plane (if it exists). The reflected line passes through both points.

If the line is parallel to the plane: reflect any point to get the image line (parallel to original, on the other side).

Projection of a line on a plane: The projection is the line in the plane closest to the original. For a line through A with direction b and plane with normal n: the projected direction is b - (b.n/|n|^2)*n (component of b perpendicular to n). The foot of perpendicular from A to the plane gives a point on the projection.