Lesson 3a - Orthogonal Trajectories

Lesson 3a - Orthogonal Trajectories -...

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Orthogonal Trajectories O t F il f C A t f d fi d b ti One parameter Family of Curves: A set of curves defined by an equation g(x,y,c)=0 where c is a parameter. That is to say an equation that only has x, y, and c. Orthogonal Trajectory: Given a one parameter family of curves g(x,y,c)=0, a relation f(x,y)=0 that intersects g(x,y,c)=0 at a right angle (regardless of the l f ) ll d h l value of c) is called an orthogonal trajectory.
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Finding Orthogonal Trajectories Ri ht l th t th l f f( ) t i i t ( ) t b Right angle means that the slope of f(x,y) at any given point (x,y) must be negative reciprocal of the slope of g(x,y,c) at the same point (x,y). This means to find an orthogonal trajectory of g(x,y,c) we must: Step 1: Derive g(x,y,c) and solve for y’. Step 2: Take the negative reciprocal of y’, this will be our new slope of our orthogonal trajectory. Step 3: Solve the differential equation y’=(negative reciprocal slope) to get your orthogonal trajectory.
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Example 1 D t i th diff ti l ti th t t th th l t j t
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