Unformatted text preview: ℓ have the same length, while projection scales the side perpendicular to ℓ —and hence the area—by a factor of cos θ . Since every triangle in P can be subdivided (using lines through the vertices parallel and perpendicular to ℓ ) into triangles of this type, the area of any triangle △ ABC is multiplied by cos θ under projection. This means V A ( △ A ′ B ′ C ′ ) = ( A cos θ ) −→ n ′ which is easily seen to be the projection of V A ( △ ABC ) onto the direction normal to the plane P ′ . We have shown Proposition 1.6.4. For any oriented triangle △ ABC and any plane P ′ in R 3 , the oriented area of the projection △ A ′ B ′ C ′ of △ ABC onto P ′ (as a...
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 Fall '08
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 Calculus, Vectors, triangle, triangle △ABC

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