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April 19th, 2019, 10:40 AM   #1
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How to obtain projected areas with direction cosines?

Hello,

I want to find a proof that shows that the projected areas of a plane with area (A) whose normal have direction cosines cos(α), cos(β), and cos(γ), on the planes XY, XZ, and YZ are given by

Az= A.cos(α)
Ay= A.cos(β)
Ax= A.cos(γ)

I found this proof but I have many Doubts about it:



Designating as c the line common to triangles A1 and A in Figure 11.1c, the areas of Az and A are cb/2 and ca/2, where b and a are the altitudes normal to c of triangles A1 and A, respectively. So Az/A = b/a

In figure 11.D we see that cos(α)=b/a. So Az/A = cos(α), and Az = A cos(α).


How do I know that the altitudes "a" and "b" meet at point c? As far as I now that is true only for pyramids whose base are regular.

How do I know if the normal that goes through the origin intersect the altitude "a"?

Is there any other valid proof?

BR

Last edited by JoseTorero; April 19th, 2019 at 10:53 AM.
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April 24th, 2019, 09:48 AM   #2
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No one?
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