3D to 2D Projection Formula: PROOF
In this video I derive the formula for projecting 3D coordinates onto a 2D screen using similar triangles. First I plot out the coordinates in 3D and then draw a straight line from the point where we are projecting the coordinates and onto the x = 0 plane. The point of projection can be thought of as the point at which our camera or eyes are located at, thus the resulting 2D projection maintains 3D perspective from that specific point. Drawing a second line passing through the z coordinates of the 3D line, we can see two similar triangles. Either of the similar triangles can be used to obtain identical formulas for the y and z projection coordinates. Note that in this projection derivation, I project the x coordinates to x = 0 and the camera or eyes location is at x = 1000. Epic stuff!
The timestamps of key parts of the video are listed below:
- Projecting 3D coordinates to 2D coordinates: 0:00
- Two Similar Triangles: 2:45
- Determining the formula for the y and z projection: 4:41
- 3D to 2D projection formula: 8:53
This video was taken from my earlier video listed below:
- Laboratory Project: Putting 3D in Perspective: https://youtu.be/3txedAqdtkQ
- HIVE video notes: https://peakd.com/hive-128780/@mes/laboratory-project-putting-3d-in-perspective
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0ElsrMs_IBprUoHocIwyAmK
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Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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