Tagged Questions
3
votes
1answer
169 views
How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?
How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$?
I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...
3
votes
3answers
113 views
How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?
Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$
My book says the answer is $(0,0),(a,0),(a,\pi)$.
However I calculated $ (a,0),(a,\pi),(a,2\pi)$.
0
votes
0answers
34 views
Curve is approximatable by function
I want to show that for $\gamma: [a,b] \subseteq \mathbb{R} \rightarrow V$ continously differentiable where V is a bounded subset of $\mathbb{R}^2$. There is always a sequence of functions ...
0
votes
0answers
30 views
Approximation theorem
I am looking for some theorem that gives me that each curve $x(t)=(x_1(t),x_2(t))$ that is continously differentiable and has $\dot{x_1}(t)\ge 0$ can be approximated by continously differentiable ...
0
votes
1answer
39 views
How To Write The equation for a line given a set of co-ordinates
I'm trying to learn how can I write equation for a line given all the points that belongs in the line.
I'm looking to find equation for a curve.
An example Set of points is:
{ (24,11) (25,11) ...
0
votes
0answers
87 views
Show that the tractrix is orthogonal to certain half-circles
Show that the tractrix discussed in Example 1.17 is orthogonal to the lower half of each circle with radius $a$ and center on the positive $y$-axis.
Example 1.17:
...
4
votes
1answer
458 views
Arc length formula for the lemniscate
This question can be homework for elementary calculus.
The lemniscate of Bernoulli $C$ is a plane curve defined as follows.
Let $a > 0$ be a real number.
Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be ...
1
vote
1answer
102 views
Finding the equation of a curve that has a perpendicular distance of $d$ from another curve
Let's say we have an equation of a curve as $y = f(x)$.
I want to find the curve $y = g(x)$ where $(x_1, f(x_1))$ has a perpendicular distance of $d$ from that curve.
Doing this with straight lines ...
1
vote
1answer
73 views
Computing the gradient of the function $\psi(u) = f[ \phi(u)]$
This question is based on section 6 of the paper Kriging and splines with derivative information.
A parametric curve $\phi(u)$ in three dimensions is deformed by the function $f$ to a new curve ...
1
vote
2answers
359 views
Parallel functions.
In 2 dimensions, we can draw 2 parallel lines that have the same distance from a line.
I wanted to find parallel functions of a function and their distance is $d$ to the function for all inputs and ...
2
votes
1answer
49 views
Equivalence of two definitions of path (in $\mathbb{R}^3$) length
In a previews question I asked here I used the following definition of path length:$\gamma=(x(t),y(t),z(t))$ : $L(\gamma)=\intop_{a}^{b}\sqrt{(x'(t))^{2}+(y'(t))^{2}+(z'(t))^{2}}$.
In the answer ...
0
votes
1answer
100 views
Prove if a polar function involves only the rational numbers and sin, cos, tan functions, it can be written in rectangular form.
Prove if a function only have including the rational numbers and sin, cos, tan function and $r$, you always could write it in rectangular form. Ex. For $r=2/(2+2\cos\theta)$ it could be represent in ...
3
votes
1answer
314 views
The area of the superellipse
I'm watching this video, where D. Knuth explains the connection of $\pi$ and factorials, and other matters (it is very interesting). Almost at the end of the talk he says the area of the superellipse
...
0
votes
1answer
342 views
Tractrix Tangent Length Problem
I am having trouble with a problem I am working on
The trace of $\vec{r}(t):=\sin(t)\vec{i}+[\cos(t)+\ln[\tan(t/2)]]\vec{j}$ where $t\in(0,\pi)$ is called a tractrix. Show the length of the line ...
0
votes
1answer
190 views
Logarithmic Spiral Calculus Question
I am working on this problem. Even some of the notation has me confused (the vectors $\vec i$ and $\vec j$).
Let $\vec r(t):=ae^{-bt}\cos(t)\vec i +ae^{-bt}\sin(t)\vec j$ where $a$ and $b$ are ...
2
votes
2answers
2k views
Polar to Parametric Equation?
I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right.
Curve C has polar equation ...
1
vote
2answers
497 views
What does “the circle is tangent to the curve” mean?
I've got a math exercise where it's said I have to prove that a circle is tangent to a curve (described by a parametric plot). Here's the graph :
So we can see that when $y=0$, the circle is really ...
11
votes
1answer
472 views
Folium of Descartes
A colleague came to me with an interesting observation:
Consider the folium of Descartes, $$x^3+y^3=3axy$$ which upon implicit differentiation of the latter yields $$\frac{\mathrm dy}{\mathrm ...
1
vote
1answer
190 views
Why is this curve convex?
I am considering the curve traced by the equation $r=a\sin 3\theta$. Specifically as $\theta$ varies from $0$ to $\frac{\pi}{6}$, $r$ varies from $0$ to $a$. How do I conclude that the curve is convex ...
1
vote
2answers
508 views
How Can I Calculate Area of Astroid Represented by Parameter?
Let $x=2\cos^3\theta$ and $y=2\sin^3\theta$ known as the astroid.
In this case, radius $r=2$.
and gray part's $x$ range is $1/\sqrt{2}\leq x\leq 2$. this deal with $0\leq\theta\leq \pi/4$.
...
4
votes
1answer
696 views
Why does the focus of a rolling parabola trace a catenary?
I keep hearing that when one rolls a parabola on a straight line, the focus traces a catenary. I kept trying to find a proof on the Internet, but no dice. How does one prove this to be true?
1
vote
1answer
216 views
rolling wheel problem
To achieve this: http://en.wikipedia.org/wiki/Square_wheel, what should $L$ be?
5
votes
1answer
490 views
Formula for curve parallel to a parabola
I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
2
votes
2answers
176 views
Looking for the name of a Rising/Falling Curve
I'm looking for a particular curve algorithm that is similar to to a bell curve/distribution, but instead of approaching zero at its ends, it stops at its length/limit. You specify the length of the ...
1
vote
1answer
193 views
Where is my (algebra) mistake? Converting parametric to Cartesian equation
I'm having a problem with my solution to a textbook exercise:
Find the Cartesian equation of the curve given by this parametric equation:
$$x = \frac{t}{2t-1}, y = \frac{t}{t+1}$$
The textbook's ...
33
votes
4answers
1k views
A circle rolls along a parabola
I'm thinking about a circle rolling along a parabola. Would this be a parametric representation?
$(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$
A gives us the radius of the circle, B changes the frequency ...
5
votes
2answers
587 views
Tractrix-like curves
Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line?
What is a parametric equation of such ...
5
votes
2answers
955 views
On deriving the arclength of a hyperbola
In my attempts to derive the closed form for the arclength of the hyperbola, I wound up with the following integral:
$$\int\frac{\sqrt{1-m\;\sin^2 u}}{\sin^2 u}\mathrm{d}u$$
I am aware that such ...
