Questions tagged [parametric]
For questions about parametric equations, their application, equivalence to other equation types and definition.
2,627 questions
2 votes
2 answers
74 views
What is the cartesian equation for the graph of the parametric curve $(x,y)= ( t^{\frac{1}{1-t}},\, t^{\frac{t}{1-t}}) $?
The parametric curve $$ \left( t^{\frac{1}{1-t}},\, t^{\frac{t}{1-t}}\right) $$ for $t \in (0,1)\cup(1,\infty)$ traces out the points on the graphs of $y = x^t$ which are furthest from the line $y = x$...
- 7,626
1 vote
0 answers
24 views
Parametric leaf profile?
I am trying to look for a nice way of parametrizing a leaf (botanical) profile. First a regular leaf, but I also would like to do an oak. The trivial answer is to use a B-Spline, but for multiple ...
- 3,857
0 votes
0 answers
83 views
Is there a trick to finding the normal function of an ellipse given its parametric definition?
I’m trying to solve a calculus problem posed like this (N is the unit normal function and T is the unit tangent function): Use the formula $\textbf{N} = \frac{d\textbf{T}/dt}{|d\textbf{T}/dt|}$ to ...
- 1
0 votes
1 answer
75 views
Is it possible to calculate the volume of a general 3D (closed & convex) parametric surface?
I have three parametric equations in two variables that give the coordinates of points on a three-dimensional, closed, convex surface. I want to find the volume enclosed by that surface, but I haven't ...
- 2,153
5 votes
1 answer
259 views
Waggle-Dance Equation 🐝
Once a foraging bee finds food, it returns to the hive and communicates the location of the food source to the colony using the elegant waggle dance. Bees interpret this dance by combining their ...
- 1,101
0 votes
1 answer
45 views
How do i modify these skewed vectors to intersect?
Essentially, my question is to modify my current equations (that are skewed for now) to set up a situation where one drone intercepts the other, to find k. where one drone is assigned as an “attack ...
- 21
1 vote
2 answers
49 views
Finding the parametric equation of a line in three dimensions
Problem: Find the parametric equation for the line through $P(-2,0,3)$ and $Q(3,5,-2)$. Answer: \begin{align*} \overrightarrow{PQ} &= ( 3 - -2 )i + (5 - 0 ) j + (-2 -3 )k \\ \overrightarrow{PQ} &...
- 4,622
-2 votes
4 answers
143 views
Determine $m \in \mathbb Z$ for which $x_1, x_2\in \mathbb Z$ [closed]
Consider the equation $(m-1)x^2-(3-m)x-m=0$ with m real numbers $m$ different from $1$, having roots $x_1, x_2$. Determine $m \in \mathbb Z$ for which $x_1, x_2\in \mathbb Z$. my ideas So I was able ...
- 522
0 votes
0 answers
24 views
Does a parametric linear feasibility program with polynomial constant term have a polynomial solution?
Let $A \in \mathbb{R}^{m \times q}$. Let $b:\mathbb R^n \rightarrow \mathbb R^{m}$ be a polynomial function homogeneous of degree 2 (i.e., $b(z) = H (z\otimes z)$, for some matrix $H$), of a variable ...
- 46
-1 votes
1 answer
95 views
How do I convert this parametric curve into an implicit curve? [closed]
I have a continuous closed parametric curve $$ \begin{align} x &= \arccos\left(-\frac{Q × \sqrt{\frac{1}{3}} \left(\tan\left(\frac{π}{4} u\right)^2 - 1\right) - \sqrt{\frac{2}{3}} \left(\tan\left(\...
- 2,153
1 vote
1 answer
57 views
Find all values of $k\in\Bbb R$ for which $(k-1)x^3 - 4x^2 + (k+2)x$ has two roots
That's problem statement: Find all values of $k\in\Bbb R$ for which the polynomial $W(x) = (k-1)x^3 - 4x^2 + (k+2)x$ has an even number of roots. We factorize by $x$, so we get $W(x) = x[(k-1)x^2 - ...
- 233
2 votes
0 answers
75 views
Every odd function has a corresponding involution
Consider functions $f$ which are involutions, i.e. \begin{align} f(f(x))=x\quad \implies \quad f'(x)f'(f(x))=1. \end{align} Under the (Legendre-like) contact transformation \begin{align} f(x)=F'(X),\ ...
- 2,506
2 votes
1 answer
208 views
Strange parametric equation of an ellipse
In his book "On the lunisolar forces which put the oceans in motion" Euler gets the following relation: $r=b+\beta \cos ^2 (\psi)$ and according to his claim (because obviously he doesn't ...
- 257
2 votes
1 answer
99 views
Following a trajectory in a double orbit system
Background first, math at the end. I am coding an animation. I have simplified it to the core problem I am trying to solve. The blue circle orbits the origin on a fixed path. The red circle orbits ...
- 123
0 votes
0 answers
29 views
Differentiating Coulomb Integrals with respect to a parameter
Suppose you have the following integral $$V_{pq}(t):=\int_{\mathbb{R}^3}\frac{\chi_p(x,t)\chi_q(x,t)}{\|x-R_c(t)\|}\mathrm{d}x,$$ where $\chi_p(x,t):= (x_1-R_{p,x_1}(t))^\ell(x_2-R_{p,x_2}(t))^m(x_3-...
- 1,023