Axis (space) transformations

This lecture shows how a 3D object (with [x,y,z] points all over) gets "flattened" (projected) on to a 2D image plane, where the object now exists as a collection of [x',y'] coordinates.

Such a "magical" projection routinely occurs in our eyes and film/movie cameras. Here we examine how it can be COMPUTED.

Onscreen

Print

The invention of (linear) perspective must have been magical for the Renaissance artists credited with it. Shown below (from http://www.cartage.org.lb/en/themes/Arts/painting/principl-tech/pers-paint/basic-pers/basicpers.htm) is a device created by Albrecht Durer, for creating perspective views of objects.

Here is a related grid-based technique:

This is an article about Brunelleschi (an architect during Renaissance) who employed perspective in his architectural renderings of buildings.

Orthographic view of buildings:

Flat view (for comparison):

Another ortho. view:

[see www.myminicity.com for many more - you can even create such cities yourself!]

Additional links (re: perspective projection):

• http://www.cs.kuleuven.ac.be/cwis/research/graphics/INFOTEC/viewing-in-3d/node8.html
• http://www.cs.virginia.edu/~gfx/Courses/2002/Intro.fall.02/Lectures/lecture10.ppt
• http://www.cs.nps.navy.mil/people/faculty/capps/iap/class2/viewing/projection.html
• http://local.wasp.uwa.edu.au/~pbourke/projection/transform/

Applet: http://www.cs.umd.edu/~mount/Indep/Connie_Peng/surface/proj.html

USGS map projections booklet: http://erg.usgs.gov/isb/pubs/MapProjections/projections.html

Off-axis (cabinet, cavalier projections): http://www.mtsu.edu/~csjudy/planeview3D/tutorial-parallel.html