Laboratory Project: Putting 3D in Perspective

In this video I go over the mathematics involved in computer graphics programming in order to clip and project a 3D image onto a 2D screen while still maintaining 3D perspective. This was taken from my Stewart Calculus book in the Laboratory Project titled Putting 3D in Perspective. Computer graphics programmers have to somehow display 3D images stored in memory onto a 2D screen for users to observe. To do this, the 3D images need to be projected from a point at which the camera or eye is viewing and then onto a 2D screen behind the object, while maintaining perspective. To do this, the equations of 4 clipping planes are first needed to be obtained to clip off the portions of image that will be beyond the projection. In this Project, I derive the clipping plane equations, as well as equations of a 3D line and 3D rectangle, and their their projected formulas. I also go over the concept of "hidden line rendering" to remove objects that are behind other 3D objects, thus saving computing power by not needing to project them onto the screen. And as always, I use the amazing GeoGebra 3D graphing calculator to plot out all the projections, and which you can play around with in this link: https://www.geogebra.org/calculator/twezjbu5

The topics covered as well as their timestamps are listed below.

• Introduction: 0:00

• Calculus Book Reference: 1:00

• Sections in Calculus Book Chapter: 1:14

• Topics to Cover: 2:02

• Laboratory Project: Putting 3D in Perspective: 2:54

  • 4 Questions: 5:03

• Question 1: 8:10

  • Equation of the 3D line: 13:00
  • Equation of the Right Clipping Plane: 18:22
  • Equation of the Top Clipping Plane: 38:49
  • Equation of the Left Clipping Plane: 53:15
  • Equation Bottom Clipping Plane: 1:04:55
  • Summary: 1:09:10

• Question 2: 1:10:54

  • Projecting 3D on to 2D: 1:12:13

• Question 3: 1:27:31

• Question 4: 1:29:39

  • Point of Intersection of the Line and Rectangle: 1:36:56
  • Point of Intersection of the Projected Line and Projected Rectangle: 2:04:43
  • Graphing it All Out in GeoGebra 3D Graphing Calculator: 2:20:10

• Outro: 2:23:31

• PDF video notes: https://1drv.ms/b/s!As32ynv0LoaIirBjY_SU7z2CXBOG1A

• 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

• Full Vectors and the Geometry of Space series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ

Related Videos:

• Vectors and the Geometry of Space: Equations of Lines and Planes: https://peakd.com/hive-128780/@mes/equations-of-lines-and-planes .

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