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MIT 18 02 - Problem Set 6 Worked Examples

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18.02A Problem Set 6 Worked ExamplesProblem 1Let F be semi-transparent plane containing the points x = 1, y = 2 and z = 3 on the threeaxes respectively. If you put your eye at the point (2, 0, 0) at what point on F will you seethink you see a point of light at (0, 1, 1)?answer:•••123PQLet P = (2, 0, 0) and Q = (0, 1, 1).We want the intersection of the line P Q and the plane F.Line: (2, 0, 0) + t(−2, 1, 1) = (2 − 2t, t, t).Plane: x +y2+z3= 1.Intersection: (2 − 2t) +t2+t3= 1 ⇒ t =67.⇒ pt. of intersection = (2/7, 6/7, 6/7).Problem 2This example will help with pset 7, not pset 6The hypocycloid is discussed on page 594-595 of the text. This is the curve formed byfollowing a point on a circle as it rolls around the inside of another circle. Similarly, anepicycloid is traced out by a point on a circle rolling around the outside of another circle.Assume:-the fixed circle has radius a and is centered at the origin.-the rolling circle has radius b.-the point tracing out the curve is called P-the rolling circle starts with its center on the positive x-axis with the point P at (a, 0).Using the hypocycloid as an example give parametric equations for the epicycloid.answer:Equating arclength along each circle we get: aθ = bφ.−−→OC = (a + b)(cos θbi + sin θbj).−−→CP = b(cos(π − (θ + φ))bi − sin(π − (θ + φ))bj)= b(− cos(θ + φ)bi − sin(θ + φ)bj).r(θ) =−−→OP =−−→OC +−−→CP= [(a + b) cos θ − b cos(θ + φ)]bi + [(a + b) sin θ − b sin(θ + φ)]bj= [(a + b) cos θ − b cos((1 + a/b)θ)]bi + [(a + b) sin θ − b sin((1 + a/b)θ)]bj⇒ x(θ) = (a + b) cos θ − b cos((1 + a/b)θ), y(θ) = (a + b) sin θ − b sin((1 + a/b)θ)18.02A Problem Set 6 Worked Examples 2xyCabP0θφθThese plots were made with Matlab using a = 1 and various values of


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MIT 18 02 - Problem Set 6 Worked Examples

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