Projections' Angle of Line in Space

问题:

Start of line (yellow) and axes are 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.

Writing code on Matlab

回答1:

You can calculate the projection angle to any plane by:

1. Obtaining the direction of the line, d = (xp - xc, yp - yc, zp - zc)

2. Normalizing d

3. Calculating the dot-product with the normal of the plane, dot(d, n) = d.x * n.x + d.y * n.y + d.z * n.z

4. Calculating the angle to the normal by a = acos(dot(d, n))

5. Finally obtaining the angle to the plane by taking b = 90 - a (assuming units in degrees - NB most math library functions use radians)

Special case: if dot(d, n) < 0, then the angle a will be greater than 90 degrees. In this case if you only want the acute angle, do b = a - 90 instead of 90 - a.

e.g. To calculate the angle to the xy plane, use n = (0, 0, 1), i.e. the z-axis, which is the normal to that plane.