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Questions tagged [curves]

For questions about or involving curves.

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2answers
94 views

Understanding Bezier Curves vs. Circular/Elliptical/Other Arcs

From what I've read, you cannot construct an elliptical or circular arc with a single bezier curve (though I read maybe you can if the arc is less than 1/4 of a circle or something small like that, or ...
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3answers
96 views

Parametrization for the figure '8' curve? [closed]

Is there a parametrization for the figure '8' curve, which is self-intersected?
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1answer
140 views

Tangent line to a 3D curve at a given 3D point

I'm trying to compute tangent vector(x,y,z) to a 3D curve(start x1,y1,z1 end x2,y2,z2) at a point (x0,y0,z0). Can anyone tell if this information is sufficient to calculate the tangent vector? If yes ...
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0answers
98 views

Circular Clothoid Curve

One clothoid curve (see figure) has the curve parameters $(C^2= 9$ x $10^8 m^2, L= 315m)$. The starting point A of this curve is the beginning point of a road. The coordinates of point A are $(2000....
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1answer
41 views

Boundary of complex bounded domain

Assume we have a bounded domain in the complex plane with smooth boundary. Is it possible to write the boundary as disjoint union of finitely/countably many closed (smooth) curves? And furthermore, ...
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0answers
52 views

Easy question about computing points of infinity

Let us consider the curve $C$ determined by $x^2t^2 + y^2t^2 + x^2y^2 - t^4$ over $F$ ( finite field with $|F|=q$ ). Maybe it is a stupid question but how can I compute the points at infinity? The ...
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1answer
113 views

how is the following curve not simple curve? [closed]

if we start from just above point and go up and end at just below point how is the curve not simple curve. ncert class 6 chapter 4 pg 72 says it is not http://ncert.nic.in/textbook/textbook.htm?femh1=...
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1answer
456 views

Parametric equation for a space curve

With reference to the following image: the blue curve has trivially a parametrization: $$(x, y, z) = (\cos\theta, \, \sin\theta, \, 0) \; \; \; \text{with} \; \theta \in [0,\,2\pi)$$ I would like ...
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3answers
37 views

Convert more complex Parametric to Cartesian Equations: $x(t)=\frac 3 2(t+\frac 1 t),y(t)=2(t-\frac 1 t)$ [closed]

Trying to these functions into a cartesian equation: $$x(t)=\frac 3 2(t+\frac 1 t),y(t)=2(t-\frac 1 t)$$
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0answers
42 views

Looking for an algorithm to create an interpolated curve function with specific requirements

I need this algorithm for an editing program to allow users to adjust curves, and I want those curves to conform to some particular requirements. For simplicity of use I would like to be able to ...
3
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2answers
502 views

Finding parametric equations of the tangent line to a curve of intersection

The question asks to find the parametric equations of the tangent line to the curve of intersection of the surface $z=2\sqrt{9-\frac{x^2}{2}-y^2}$ and the plane $x=2$ at the point $(2,\sqrt{3},4)$ I'...
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1answer
84 views

How do I find the radii for the fixed circle and rolling circle to represent an epicycle as an epitrochoid?

Suppose we have a moon $M$ following a circular orbit (with radius $R_M$) around a planet $P$, which is in turn following a circular orbit (with radius $R_P$) around a star $S$. The angular velocity ...
2
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1answer
40 views

Proof that the average of a convex curve is inside the curve.

Let $\boldsymbol{\gamma}(t): [a,b] \rightarrow \mathbb{R}^2$ be closed curve in the plane. The region inside the closed curve is $D$, therefore $\boldsymbol{\gamma} = \partial D$ (boundary of $D$). ...
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2answers
506 views

Curvature of curve: equivalence between tangent vector and angle definitions

I know that curvature for some curve $C$ defined parametrically is: $$\kappa=\left\|{d\vec{T}\over ds}\right\|$$ Which basically is the rate at which the tangent vector to the curve changes, as the ...
0
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1answer
52 views

Linear regression analysis where the fit values must be greater than the observed values?

Long story short, I would like to efficiently:Minimize ||bX-y||2 subject to X ≥ 0 and bX ≥ y I have an observation that is a single curve (y) in the form of signal intensity vs. frequency. I ...
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2answers
48 views

Fitting splines to famous curves programmatically

this may be a bit of a naive question. I am looking for a way to input a Cartesian description of a famous curve and map that to some spline say NURBS to make spline paths. Is this at all possible? ...
0
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1answer
86 views

The set of orthonormal frames along a curve

Suppose $\gamma \colon [0,\alpha] \to\mathbb{R}^{3}$ is a smooth unit-speed curve. Let $(E,V,W)$ be a $\gamma$-adapted orthonormal frame along $\gamma$ (by $\gamma$-adapted I mean $E=\dot{\gamma}$). ...
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2answers
91 views

How to show an equation gives a closed curve

I am analyzing the following dynamical system: $\ddot{x} + x +\epsilon x^3 = 0$, which I have rewritten to 2D system $\dot{x} = y, \dot{y} = -x-\epsilon x^3$. I have found that this is a conservative ...
1
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1answer
56 views

Affine plane curves with constant curvature

Question I want to solve this differential equation for $P : \mathbb{R} \to \mathbb{A}^2$, a plane affine curve. $ P'''(t) = \frac{P'(t)}{t^2}$ Someone recognize this equation? Is a famous curve? ...
4
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2answers
54 views

Does the curve $2^2 x^2 + 4^2 y^2 = (x^2 + y^2)^2$ pass through $(0,0)$?

Any ideas about why Desmos/Wolfram Alpha) does not show $(0,0)$ as a part of the curve $$2^2 x^2 + 4^2 y^2 = (x^2 + y^2)^2$$ will be appreciated. Am I missing something or is it a bug?
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1answer
43 views

Visualizing double points.

I was trying to visualize by drawing a curve / figure to get a double point on a curve. As per the Wolfram article, a double point is a point traced out twice as a closed curve is traversed. Any ...
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0answers
19 views

How does one go about creating simple approximations of curves of known values (with tolerances)?

I'm currently working on a synth, and many of the knobs are not linear in nature. Often, knobs will have a simple algorithm by which they transform linear changes into curved ones (such as volume or ...
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1answer
51 views

Moving plane over a fixed plane

I am reading the following work https://www.mathcurve.com/courbes2d.gb/base/base.shtml and I need some explanation about the terminology. There is a problem with respect to the definition of the ...
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0answers
35 views

Directed curvature of a curve

So I have this following exercise Consider the curve given by the graph of the sine function $t \rightarrow (t,\sin(t))$. Determine the directed curvature at each point of this curve. Supposing that ...
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1answer
63 views

Plane Curve Examples

teaching myself about plane curves, both affine and projective, and I was hoping to gain some exposition via some examples, if anyone can help me out. First, I am thinking about irreducible plane ...
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0answers
50 views

computes the $\phi$ angle of an ellipse

I've an ellipse that is rotated and not centered in the origin. Hence, I've an ellipse equation that is in the form of: \begin{equation} \frac{((x-h)cos(\phi) + (y-k)sin(\phi))^2}{a^2} + \frac{((x-h)...
1
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1answer
101 views

Space filling curve

Let $C$ be the Cantor set. Then the Cantor function $f:C \to [0,1]$ can be extended to $F:[0,1]\to [0,1]$ linearly as the end points of an removed interval takes the same value. For example $f(1/3)=f(...
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0answers
35 views

Space filling curves: Hilbert vs Peano

It seems that both the Peano's and Hilbert's space filling curve are same in nature. What is the basic difference between them?
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0answers
33 views

Some doubt regarding space filling curve.

Peano's idea was to define a sequence of curves that visit ever more points than the previous. Hence the limiting curve will visit a dense set of points in the square. My doubt: we need to show that ...
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2answers
48 views

Looking for a sine curve with custom midpoint

For a signal generating software, I'm looking for a function that generates a sine-based curve but with a shifted midpoint. Usually, a sine curve looks like this: ...
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2answers
130 views

Drawing De Boor's algorithm

Assume we have 4 control points $[c_0, c_1, c_2, c_3]$ and uniform knot sequence $[0,1,2,3]$ If we were to draw an quadratic bezier we would be forced to use only 3 of the control points and then we ...
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0answers
59 views

Angle of the slope of an ellipse

Find the relation between the angle made by a straight line connecting the origin and an ellipse (angle made between that line and the x axis) and the slope of the tangent to the ellipse. This should ...
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1answer
28 views

Find Equations of tangent lines

I'm having a hard time figuring this out. I'm asked to find the the equations of the horizontal lines to the curve of $$y=x^3-3x+1$$ I set the derivative equal to zero and solve for x, to find the ...
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1answer
96 views

Confusing myself with Cox de boor's algorithm

I am using the recursive definition to understand the algorithm, mainly: $$ B_{i,0}(t) = \begin{cases} 1 & \text{if} & t_i \leq t < t_{i+1} \\ 0 & \text{otherwise} \end{...
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0answers
34 views

Show that these curve families are orthogonal: $f(xy) = C$ and $y^2 - x^2 = D$ [duplicate]

Let the function $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable and its derivative is never zero. Show that these curves are orthogonal: $$f(xy) = C$$ $$ y^2 -x^2 = D$$ $C$ and $D$ are ...
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1answer
48 views

Parametric equation of a non-wrapping circle on the surface of a cylinder

I have a cylinder of radius $R$, and I wish to draw a circle of radius $r$ on its surface that does not wrap around it. So, the centre will lie somewhere on the surface of the cylinder, and the whole ...
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0answers
42 views

General properties of closed curves on a sphere

This is a question from a physicist trying to understand space curves. In 2D, a closed curve equidistant from a point is a circle, and has constant curvature and zero torsion. What can be said for ...
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1answer
28 views

Proof of equivalence well defined function

This is the definition: Definition 1: A function $f:D\subseteq R\to R^n$ is said to be continuously differentiable of a $C^1$ function, if f is differentiable and the first derivative of f is ...
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1answer
27 views

find the equation of a 'level' curve, for a function $\mu(x_1,x_2)=0$

I would like to ask the following question. I have an aggregated function, $$ \mu(x_1,x_2) = \|\nabla f_2\| f_1(\vec{x}) + \|\nabla f_1\| f_2(\vec{x}), $$ where the norm of the gradients are also ...
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0answers
29 views

Smoothly merging two parametric curves

Let's imagine that an object follows a path described by the known parametric curve $t(s)$ for $s \geq 0$. Now, another object follows another curve $c(s)$, that goes through a known point $c_0$. I ...
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0answers
17 views

Function for providing coordinates of next point along a pseudo Hilbert curve of a particular order?

It's easy to write a recursive program to draw an n'th order pseudo Hilbert curve, but I am interested to know if there is an efficient linear-time solution (without an iterative or recursive program) ...
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0answers
61 views

Smoothing and normalizing a singular curve

If we have a singular curve for example nodal curve then we can define its normalization. I read somewhere we have also smoothing a singular curve Does anyone know the definition or any references?
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2answers
59 views

“fastest” curve through n points

I'm programming an AI for a race game, where my car has to drive through some checkpoints. If I let it drive straight in direction of the checkpoints, it has to slow down and make a huge turn after ...
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0answers
19 views

Prove that the following function is an orientation of a curve

Assume that $f^{(j)}:\mathbb{R}^3\rightarrow \mathbb{R}$ for $j=1,2$ are some $C^1$ functions such that $(\nabla f^{(1)}, \nabla f^{(2)})$ are independent over the curve $\gamma = \{ x \in \mathbb{R}^...
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0answers
32 views

From wavelets to curvelets

I've read that the curvelet is a generalisation of wavelet. I am now looking for references such as books, lecture note or research papers introducing the mathematical theoretic aspect of curvelet to ...
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1answer
28 views

Cool Curves to rotate about axis

I am working on a project for my Calculus class in which I have to rotate a curve about some axis and 3-d print a model of the curve (using any type of cross-sectional areas). At first, I tried to ...
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0answers
25 views

Piecewise linear robot motion (with obstacles)

The problem is best described with the image below. An robot (the diamond shape) is only allowed to move in a piecewise linear fashion, with each halt being a point of a curve (the red curve). The ...
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1answer
30 views

Equation describing the deformation of a beam with large displacement

Short version of the question Which is the equation describing the shape of a beam with large displacement knowing the starting and ending points and tangents? My considerations: The Eulero-...
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0answers
16 views

Length of figures traced by zeroes of a complex function

If I have a complex function $f$ whose zeroes make curves in the complex plane, is there a way of getting the possibly-infinite length of all the curves that $f$ traces out? In particular, is there a ...
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0answers
12 views

For which values of $x$ we will get that these curves are orthogonal?

For which values of $x$ we will get that: $$ x^2 + cy^2 = 7 $$ (while $c>0$) orthogonal to: $$ x^2 + y^2 = log(x^{14}) + 29 - 14\cdot log(2) $$ ? My intuition was that it's while $x>0$ but it's ...