Sketch the curve . At the point , label and . Give the values of and at this point as well.
Problem 2
Consider the circle . Find , , and .
Problem 3
The function , , defines a spiral centred at the origin. Using only geometric intuition (no calculation), predict .
As the local effect of the curve is that it becomes more and more flat at any point via the term, meaning it effectively becomes a line which has a curvature of 0 so .
Problem 4
Let . What is ?
Problem 5
Consider that the spiral parametrized in terms of arc length is . Find and for this curve.
Problem 6
Given a curve , compute the following quantities:
(A).
(B).
(C).
(D).
(E).
Problem 7
Find the curvature of .
Problem 8
Find the minimum and maximum values for the curvature of the ellipse , . Here .
The maximum occurs at and the minimum occurs at because
Problem 9
Consider the curve .
(a). Find the Curvature at
(b). Find the Equation of the Circle Best Fitting at
Problem 10
Consider the motion of a thumbtack stuck in the tread of a tyre which is on a bicycle moving at constant speed. This motion is given by the parametrized curve $$\vec{r}(t) = \big(t - \sin t, 1 - \cos t\big)$$
with .
(a). Find and Simplify the Formula for the Curvature
(b). Find the Radius of Curvature of the Osculating Circle to at
(c). Find the Equation of the Osculating Circle to at
Problem 11
Find the curvature as a function of arc length (measured from ) for the curve $$x(\theta) = \int_0^\theta \cos\left(\tfrac{1}{2}\pi t^2\right)dt, \qquad y(\theta) = \int_0^\theta \sin\left(\tfrac{1}{2}\pi t^2\right)dt$$