math - Must polygon prism have interior angles >144 degree to have sides visible after projection transform? -


most view frustums 35 45 degrees, angle of each of 4 sides slope near plane far plane. these exterior angles. 36 degree frustum has interior angles of 144 degrees. projection transform generates box, rather frustum. sides 144 degree interior angles swivel in 90 degrees.

now consider 10-sided prism, decagon prism, has same angles frustum. if viewer sees 1 of faces orthogonally, flat 2d surface, it's neighboring faces virtually disappear after projection transform, reduced 90 degree angles.

am correct or wrong?

am correct or wrong?

wrong (see below explanation why).

these exterior angles.

field of view defined internal angle. isn't important general point of question, let's ignore it, means fov angle specified 1 way instead of another.

now consider 10-sided prism, decagon prism, has same angles frustum. if viewer sees 1 of faces orthogonally, flat 2d surface, it's neighboring faces virtually disappear after projection transform, reduced 90 degree angles.

the hidden assumption here determines visibility of face is:

  • the angle of face
  • the angle viewer facing, and
  • the viewer's field of view.

this seems reasonable on surface, it's wrong. of these things influence whether face visible viewer, don't determine on own. what's missing equation viewer's position.

what actually determines whether face visible viewer whether viewer's eye position lies on correct side of plane face lies on, extended in directions infinity. secondarily, face must in viewer's field of view.

to convince of this, stand in front of door opens towards you, , open 60 degrees. take couple of steps back. should able see side of door faces room standing in. walk forward through door, facing forward whole time. @ point, side of door see become invisible, , other side of door become visible. obviously, angle of door hasn't changed, , neither has direction facing, has caused change in visibility not angles fact have passed 1 side of plane door lies on other.

here's diagram illustrate in case of decagon:

enter image description here

the diagram shows top down view. red line represents plane of 1 of faces of decagon. blue dots represent different viewer positions , blue lines represent field of view angles. when viewer's eye position on "can't see face here" side of plane, face not visible, , vice versa other side.

for viewer "sees 1 of faces orthogonally, flat 2d surface" (the front face), able see face @ position 3 (indicated green line), not @ position 2 (because of narrow fov), , not @ position 1 (because of being on wrong side of plane face lies on).


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