Projecting 3D points to 2D plane [closed]

by

If you have your target point P with coordinates r_P = (x,y,z) and a plane with normal n=(nx,ny,nz) you need to define an origin on the plane, as well as two orthogonal directions for x and y. For example if your origin is at r_O = (ox, oy, oz) and your two coordinate axis in the plane are defined by e_1 = (ex_1,ey_1,ez_1), e_2 = (ex_2,ey_2,ez_2) then orthogonality has that Dot(n,e_1)=0, Dot(n,e_2)=0, Dot(e_1,e_2)=0 (vector dot product). Note that all the direction vectors should be normalized (magnitude should be one).

Your target point P must obey the equation:

r_P = r_O + t_1*e_1 + t_2*e_2 + s*n

where t_1 and t_2 are your 2D coordinates along e_1 and e_2 and s the normal separation (distance) between the plane and the point.

There scalars are found by projections:

s = Dot(n, r_P-r_O)
t_1 = Dot(e_1, r_P-r_O)    
t_2 = Dot(e_2, r_P-r_O)

Example with a plane origin r_O = (-1,3,1) and normal:

n = r_O/|r_O| = (-1/√11, 3/√11, 1/√11)

You have to pick orthogonal directions for the 2D coordinates, for example:

e_1 = (1/√2, 0 ,1/√2)
e_2 = (-3/√22, -2/√22, 3/√22)

such that Dot(n,e_1) = 0 and Dot(n,e_2) = 0 and Dot(e_1, e_2) = 0.

The 2D coordinates of a point P r_P=(1,7,-3) are:

t_1 = Dot(e_1, r_P-r_O) = ( 1/√2,0,1/√2)·( (1,7,-3)-(-1,3,1) ) =  -√2
t_2 = Dot(e_2, r_P-r_O) = (-3/√22, -2/√22, 3/√22)·( (1,7,-3)-(-1,3,1) ) = -26/√22

and the out of plane separation:

s = Dot(n, r_P-r_O) = 6/√11

More Related Contents:

  1. A simple algorithm for polygon intersection
  2. Turning a cylinder into a sphere without pinching at the poles
  3. Signed angle between two 3D vectors with same origin within the same plane
  4. Perpendicular on a line from a given point
  5. how map 2d grid points (x,y) onto sphere as 3d points (x,y,z)
  6. Calculate distance between 2 GPS coordinates
  7. Generate a random point within a circle (uniformly)
  8. How to determine if a list of polygon points are in clockwise order?
  9. How to calculate the volume of a 3D mesh object the surface of which is made up triangles
  10. Mathematically compute a simple graphics pipeline
  11. Cone to box collision
  12. How do I calculate the normal vector of a line segment? [closed]
  13. Calculating area enclosed by arbitrary polygon on Earth’s surface
  14. How do I combine complex polygons?
  15. correcting fisheye distortion programmatically
  16. Implementing Ray Picking
  17. Rotating a Vector in 3D Space
  18. Calculating a LookAt matrix
  19. How do I detect intersections between a circle and any other circle in the same plane?
  20. How do I reverse-project 2D points into 3D?
  21. How to distribute points evenly on the surface of hyperspheres in higher dimensions?
  22. How to calculate Tangent and Binormal?
  23. Plotting a point on the edge of a sphere
  24. Finding the coordinates of points from distance matrix
  25. 3D coordinates on a sphere to Latitude and Longitude
  26. Calculate rotated rectangle size from known bounding box coordinates
  27. Line of intersection between two planes
  28. Flipping a quaternion from right to left handed coordinates
  29. How to best approximate a geometrical arc with a Bezier curve?
  30. Get bounds of rotated rectangle

Leave a Comment