⋆ If the n(n + 3 )/ 2 points determining an n th degree curve are chosen to be among the n 2 points that two n th degree curves share, there seems to be a lack of determinacy. ⋆ Resolution: The equations aren’t independent. Descriptive Geometry • What is it? ◦ The study of the representation of 3-dimensional objects using 2-dimensional Fgures. ◦ think of mechanical drawings, drafting, etc. • Essentially invented by Monge in the late 1700s. (Much geometry done in revolutionary ±rance. Monge’s motivation: To “pull the ±rench nation out of its hitherto dependence on foreign industry”.) • Uses parallel projection. • Monge went on to do nontrivial mathematics with such representations. Projective Geometry • What is it? ◦ The study of geometric properties of and classes of geometric objects that are un-changed by projections, or changes of perspective. ◦ Original motivation was a desire to help painters: A painting should be a section of the projection of the lines of light from the thing being painted to the eye. ◦ Projection preserves neither congruence, nor similarity, nor angle, nor length, nor area.
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