Chapter 3: Applications of Differentiation
Section 3.1: Tangent and Normal Lines
Let C1 be the ellipse defined by x=uθ=5 cosθ,y=vθ=3 sinθ,0≤θ≤2 π, and let C2 be the curve defined parametrically by x=Uθ=165cos3θ,y=Vθ=−163sin3θ,0≤θ≤2 π.
Graph C1 and C2 on the same set of axes.
Show that for each value of θ, the slope of the line normal to C1 is the same as the slope of the line tangent to C2.
In the animation in Figure 3.1.3(a), the ellipse C1 is drawn in red; the curve C2, in black; and the line normal to C1, in green. Note that each such normal line is actually tangent to C2.
The curve C2 is called the evolute of C1, and C1 is called the involute of C2.
The evolute of a plane curve can be characterized as the envelope of the family of lines normal to the involute.
The evolute is also the locus of the centers of curvature of the involute. See Section 3.5 for a discussion of curvature and circles of curvature.
Define curves C1 and C2
Write the equations uθ=… and vθ=…
Context Panel: Assign Function
uθ=5 cosθ→assign as functionu
vθ=3 sinθ→assign as functionv
Write the equations Uθ=… and Vθ=…
Uθ=165cos3θ→assign as functionU
Vθ=−163sin3θ→assign as functionV
Write the negative reciprocal of the slope along C1.
Context Panel: Evaluate and Display Inline
−1v′θu′θ = 53⁢sin⁡θcos⁡θ
Write the expression for the slope along C2.
V′θU′θ = 53⁢sin⁡θcos⁡θ
Write the list uθ,vθ.
Context Panel: Plot Builder≻2-D plot (parametric)
Write the list Uθ,Vθ.
The graph of C1 can be superimposed on the graph of C2 by copying one and pasting it onto the other.
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