Chapter 3: Applications of Differentiation
Section 3.5: Curvature of a Plane Curve
Example 3.5.2
Show that the circle x−h2+y−k2=r2 everywhere has constant curvature, that is, show κ=1/r.
Solution
Mathematical Solution
Implicitly differentiate the equation of the circle to obtain y′:
x−h2+y−k2
=r2
2x−h+2y−ky′
=0
y′
=−x−hy−k
Obtain and simplify the denominator of κ:
1+y′23/2
=1+−x−hy−k23/2
=y−k2+x−h2y−k23/2
=r2y−k23/2
=r3y−k3
The final simplification hinges on the positivity of r. Note also that the square root of y−k2 is y−k, which matters because y−k can be both positive and negative along the circle.
Implicitly differentiate y′ to obtain y″, the numerator of κ:
y″
=ddx−x−hy−k
=y−k−x−hy′y−k2
=y−k−x−h−x−hy−ky−k2
=y−k2+x−h2y−k3
=r2y−k3
Finally, obtain κ=|y″|1+y′3/2=r2y−k3r3|y−k|3=1r.
Maple Solution
Initialize
Control-drag the equation of the circle. Context Panel: Assign to a Name: C
x−h2+y−k2=r2→assign to a nameC
Obtain y′
Type the name C and Press the Enter key.
Context Panel: Differentiate≻Implicitly In the dialog that appears (see Figure 3.5.2(a)), set y as the dependent variable, remove y from the list of independent variables, and set x as the variable of differentiation.
Press the OK button in the dialog.
Figure 3.5.2(a)
C
x−h2+y−k2=r2
→implicit differentiation
−x+hy−k
Obtain y″
Type the name C and press the Enter key.
Context Panel: Differentiate≻Implicitly Set y as the dependent variable, remove y from the list of independent variables, and write x,x for "Differentiate with respect to".
−h2−2⁢x⁢h+k2−2⁢y⁢k+x2+y2−k3+3⁢y⁢k2−3⁢y2⁢k+y3
Form and simplify κ=|y″|1+y′23/2
Using equation labels, write the expression for κ, then press the Enter key.
Context Panel: Simplify≻Assuming Real
Context Panel: Simplify≻With Side Relations Enter C for Relations in the Specify Relations dialog, then click the OK button.
Control Panel: Simplify≻Assuming Positive
1+23/2
h2−2⁢x⁢h+k2−2⁢y⁢k+x2+y2k3−3⁢y⁢k2+3⁢y2⁢k−y31+−x+h2y−k232
→assuming real
1h2−2⁢x⁢h+k2−2⁢y⁢k+x2+y2
= simplify siderels
1r2
→assuming positive
1r
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