Problem 1
In the sketch below of a three-dimensional curve and its osculating circle at a point, label and . Will be pointing out of the paper towards the reader, or into the paper away from the reader?
Via the right hand rule will be pointing out of the paper.
Problem 2
In the formula $$\frac{ds}{dt}(t) = |\mathbf{v}(t)| = |\mathbf{r}'(t)|$$
does stand for speed, or for arc length?
stands for the arc length as speed is which doesn't fit with it's derivative wrt being equal to speed.
Problem 3
Which curve (or curves) below have positive torsion, which have negative torsion, and which have zero torsion? The arrows indicate the direction of increasing .
For and the torsion is negative and for its 0.
Problem 4
Consider a curve that is parametrized by arc length .
(a). Show that if the Curve Has Curvature for All , then the Curve is a Straight line
The curvature describes how much a curve "bends" away from it's straight line so if it's zero the curve must be a line as it never bends away from a line.
(b). Show that if the Curve Has Curvature and Torsion for All , then the Curve Lies in a Plane
The torsion describes how much the curve rotates out of the osculating plane, if it's 0 it's always within the same osculating plane and never rotates out of it.
(c). Show that if the Curve Has Curvature , a Strictly Positive Constant, and Torsion for All , then the Curve is a Circle
If we have a constant curvature the curve must "bend" the same amount at every point additionally because the torsion is zero we lie on a plane meaning the curve can only bend in one dimension so it must bend in a circle.
Problem 7
Find the torsion of at the point .
Problem 8
Find the unit tangent, unit normal and binormal vectors and the curvature and torsion of the curve $$\mathbf{r}(t) = t,\hat{\mathbf{i}} + \frac{t^2}{2},\hat{\mathbf{j}} + \frac{t^3}{3},\hat{\mathbf{k}}$$
Problem 9
For some constant , define . For which value(s) of is ? For each of those values of , find an equation for the plane containing the osculating circle to the curve at .
Problem 10
(a). Consider the Parametrized Space Curve . Find an Equation for the Plane Passing through with Normal Vector Tangent to at that point
(b). Find the Curvature of the Curve from (a) as a Function of the Parameter
Problem 11
Let be the osculating circle to the helix at the point where . Find:
(a). The Radius of Curvature of
(b). The Centre of
(c). The Unit Normal to the Plane of
Problem 12
(b). Find the Tangential Component of Acceleration, as a Function of , for the Parametrized Space Curve
Problem 13
Suppose, in terms of the time parameter , a particle moves along the path $$\mathbf{r}(t) = (\sin t - t\cos t),\hat{\mathbf{i}} + (\cos t + t\sin t),\hat{\mathbf{j}} + t^2,\hat{\mathbf{k}}, \qquad 1 \le t < \infty$$
(a). Find the Speed of the Particle at time
(b). Find the Tangential Component of Acceleration at time
(c). Find the Normal Component of Acceleration at time
(d). Find the Curvature of the Path at time
Problem 14
Assume the paraboloid and the plane intersect in a curve . is traversed counter-clockwise if viewed from the positive -axis.
(a). Parametrize the Curve
(b). Find the Unit Tangent Vector , the Principal Normal Vector , the Binormal Vector and the Curvature , All at the point
Problem 15
Consider the curve given by $$\mathbf{r}(t) = \frac{1}{3}t^3,\hat{\mathbf{i}} + \frac{1}{\sqrt{2}}t^2,\hat{\mathbf{j}} + t,\hat{\mathbf{k}}, \qquad -\infty < t < \infty$$
(a). Find the Unit Tangent as a Function of
(b). Find the Curvature as a Function of
(c). Determine the Principal Normal Vector at the point
Problem 16
Suppose the curve is the intersection of the cylinder with the plane .
(a). Find a Parametrization of
(b). Determine the Curvature of
(c). Find the Points at Which the Curvature is Maximum and Determine the Value of the Curvature at These Points
Problem 17
Let . Compute the unit tangent and unit normal vectors and . Compute the curvature . Simplify whenever possible.
Problem 18
(a). Find the Length of the Curve for
(b). Find the Principal Unit Normal Vector to at
(c). Find the Curvature of at
Problem 19
A particle moves along a curve with position vector given by $$\mathbf{r}(t) = \big(t + 2,; 1 - t,; t^2/2\big), \qquad -\infty < t < \infty$$
(a). Find the Velocity as a Function of
(b). Find the Speed as a Function of
(c). Find the Acceleration as a Function of
(d). Find the Curvature as a Function of
(f). Find an Equation for the Osculating Plane (the Plane Which Best Fits the curve) at the point Corresponding to
(g). Find the Centre of the Osculating Circle at the point Corresponding to
Problem 20
Consider the curve given by $$\mathbf{r}(t) = \frac{t^3}{3},\hat{\mathbf{i}} + \frac{t^2}{\sqrt{2}},\hat{\mathbf{j}} + t,\hat{\mathbf{k}}, \qquad -\infty < t < \infty$$
(a). Find the Unit Tangent as a Function of
(b). Find the Curvature as a Function of
(c). Evaluate at
(d). Determine the Principal Normal Vector at
(e). Compute the Binormal Vector at
Problem 21
A curve in is given by .
(a). Find the Parametric Equations of the Tangent line to the Curve at the point
(b). Find an Equation for the Osculating Plane of the Curve at the point
Problem 22
A curve in is given by $$\mathbf{r}(t) = (\sin t - t\cos t),\hat{\mathbf{i}} + (\cos t + t\sin t),\hat{\mathbf{j}} + t^2,\hat{\mathbf{k}}, \qquad 0 \le t < \infty$$
(a). Find the Length of the Curve from to
(b). Find the Curvature of the Curve at time