# Questions tagged [parametric]

For questions about parametric equations, their application, equivalence to other equation types and definition.

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### Convert Parametric Polynomial Equation to Cartesian - Always Possible?

Let $p(t),q(t)\in\Bbb R[t]$. Do there always exist nonconstant $f(t),g(t)\in\Bbb R[t]$ such that $f(p(t))=g(q(t))$? I started by looking at a tricky example $$x=t^5+2t^3+3t, y=t^6+t^4+t^2$$where there ...
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1 vote
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### Significance of tangent indicatrix

I'm reading a proof of the Four Vertex Theorem, and the proof introduces the notion of a tangent indicatrix. The precise text is as shown above. Can someone explain the intuition behind the tangent ...
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### What is this formula? Area of Jordan curve

I came across the following area formula in do Carmo's book: I tried searching online for where this formula came from, but I couldn't find anything that matched this. Does anyone have a name for the ...
• 111
1 vote
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### Parametric Equation of a unit circle when the angle between $x$-axis and $y$-axis is not $90$ degrees

I know in regular Cartesian coordinates the parametric equation for a unit circle is $x=\cos(\theta)$, $y=\sin(\theta)$, and if the $x$ coordinates are stretched by an amount $a$, and the $y$ ...
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### Parameterization by arc length: the concept

Can someone let me know if my understanding of parameterization by arc length is correct? If we have a regular parameterized differentiable curve $\alpha: I \rightarrow \mathbb{R}^3$, it is ...
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1 vote
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### Determining the Equation of a Parabola in a Three-Dimensional Plane

I have a plane with three points: P1 (x1, y1, z1), P2 (x2, y2, z2) and P3(x3, y3, z3). These points represent a parabola where P3 is the vertex and the other two points are the ends of the parabola. ...
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### Calculating Slip Between Two Curves at Their Contact Point in Space

Question: Calculating Slip Between Two Curves at Their Contact Point in Space I want to calculate the slip between two curves at their contact point in space as they move and/or stretch. The curves ...
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### How can we draw $x(t)=3\sin(t) \sin(3t),y(t)=3\cos(t) \cos(3t)$? [closed]

I tried to take multiple values for $t$ but this was a painful procedure. I wondered if there is any way to make to graph $(x,y)$ faster. However, every way I tried failed to work. For example, ...
33 views

### Number of solutions in the positive octant $\mathbb{R}^3_+$ of a polynomial system of equations

Important edit: apparently my conjecture was wrong (see the answer I posted). In any case, I would still be very interested in methods that allows me to find/estimate the number of solutions in the ...
23 views

### Determine the Domain Extents of an Arbitrary Bounded Cylinder in Cylindrical Coordinates

More Description I'm trying to determine the domain extents of a cylinder with an arbitrary orientation and size in cylindrical coordinates. I need the domain extents, or the maximum and minimum ...
1 vote
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### Is there a way to derive the Polar Curve Area Formula using Parametrics?

I just finished up Calc BC, and one formula that my teacher never really went into the derivation of was how the area of a polar curve is given by $A=\frac12\int_\alpha^\beta r^2d\theta$. One that I ...
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### Converting a set of parametric coordinates to Cartesian coordinates

I have the parametric set of coodinates P = $(\dfrac{1+at^{2}+6at}{4},\dfrac{3at^{2}+2at+3}{4})$ , where $a = \dfrac{1}{16}$, after solving a problem involving the locus of a point. I wish to convert ...
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1 vote
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### How to modify off-center circle in polar coordinates so that input angle has a linear relationship with angle on circle?

I have a circle translated horizontally in polar coordinates described by the equation: $$r\left(\theta\right)=d\cos(\theta)+\sqrt{r_{0}^{2}-d^{2}\sin^{2}(\theta)}$$ where $d$ is the horizontal ...
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1 vote
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### Number of solutions of parametric equation $\sin\left(a(\sin x+\cos^2x)\right)=0$ for $a\in\mathbb{R}^+$ and $0\leq x\leq\frac{\pi}{2}$

I am trying to find the number of solutions $N(a)$ of the following parametric equation: $$\sin\left(a(\sin x+\cos^2x)\right)=0,$$ where $a\in\mathbb{R}^+$ and $0\leq x\leq\frac{\pi}{2}.$ What I have ...
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### Insights for Outcome Function Involving Multiple Interdependent Variables

I am working on a model involving multiple interdependent variables and systems of equations, and I am trying to gain insights into the behavior and properties of a specific outcome function. Despite ...
1 vote
76 views

### Convert pair of parametric trig equations to $y=f(x)$ form

My apologies if this already has an answer, I've spent some time looking but haven't found anything that (to me) looked directly applicable. I have a set of parametric equations describing a periodic ...
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84 views

### Finding area using green's theorem

A friend of mine gave me a variant of the goat problem, which is the following: If a goat is tied to a circular fence of radius $10$ feet with a rope of $20$ feet, how much land can the goat roam? I ...
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### Writing the equation of a family of circles touching two circles

I know that the equation of a circle passing through the intersection of two circles is $S_1+\lambda S_2=0$ and I know that the equation of a circle passing through the intersection points of a line ...
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