Projections' Angles of Line in Space

Start of line (yellow) and axes at [xc,yc,zc]

End of line is at [xp,yp,zc].

a,b,c are the angles which line makes in space.

What I need are the angles which line's projections (black line) create on xy,yz and xz planes.

• A_y_to_z: Projected line's angle from y axis to z axis on xz plane.
• A_z_to_x: angle from z to x axis on zx plane.
• A_x_to_y: Angle from x to y axis on xy plane.

Answer 1

[Jan]

The vector [xp, yp, zp] projected in the XZ plane is [xp, 0, zp]. Its normal vector is:

[xp, 0, zp] / norm([xp, 0, zp]);

The angle can be determined by the dot product with [0,0,1]:

A_ztox = acos(zP / sqrt(yp^2 + zp^2));

The same for the other two angles:

A_xtoy = acos(xP / sqrt(xp^2 + yp^2));
A_ytoz = acos(yP / sqrt(yp^2 + zp^2));

See https://en.wikipedia.org/wiki/Direction_cosine and set one component to zero.