# Tagged Questions

Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes also the image $\gamma(I)$ is called curve.

20 views

### Intersection Multiplicity of Rational Plane Curves

Suppose I have two rational curves in the complex projective plane. I know their parametrizations, $<x_1(t),y_1(t)>$ and $<x_2(t),y_2(t)>$ I know I can use Grobner bases to find an ...
63 views

### How do you find the angle of intersection between two given polar curves?

How does one find the angle of intersection between two given polar curves? For example, between $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
20 views

### Intersection point of the normal lines at α(t), α(t+h) converges as h→∞ given α is parametrized by arc lenght and it´s curvature is non zero

Let α(t):I→R2 be a curve parametrized by arc lenght and k(t) (curvaure) be non zero. Need to show the intersection point of the normal lines at α(t), α(t+h) converges to a point in the trace of the ...
51 views

### Does every graph have an algebraic form?

Let's say I take a pencil and start drawing a curve on $xy$ plane. The curve is continuous and for each value of $x$ there is only one corresponding value of $y$. So question that interests me is - ...
67 views

### Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
190 views

### Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...
39 views

### Vector analysis: understanding formulas for normal and tangent

I'm studying a chapter on vector analysis and I don't really understand why the following equations are switched for the two situations. (My course is not in English, so I apologize if I translated ...
42 views

### Set-theoretic equality in double dual graph

Someone in the math fandom on tumblr was explaining the concept of a dual graph, and he was asked a tough question: Is the dual of the dual [equal to] the original graph always, or just isomorphic?...
42 views

161 views

### What is the circumference (arc length) of $x^4 + x^2 + y^4 + y^2 = 2$?

Consider $$x^4 + x^2 + y^4 + y^2 = 2$$ It is a smooth non-intersecting circle like curve in the plane. A bit like a Hyperellipse. See https://en.m.wikipedia.org/wiki/Superellipse What is the ...
70 views

### Genus-Degree formula gives the wrong answer: ordinary points?

I'm trying to compute the genus of the normalization of the curve: $y^5=x(x-1)(x-2)$ Now I calculate the ramification points of the projection x: they are $(0,0),(1,0),(2,0)$ and they are of ...
47 views

### Circle of radius of Intersection of Plane and Sphere

The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius? I tried putting value of y from plane in sphere but then I get a $zx$ term. How to proceed?
25 views

22 views

### Basic plane question, finding a plane traveling through the heads of 3 given vectors.

Find the equation for the plane passing through the heads of the three given vectors (2, 2, 0) (−1, 2, 1) (1, 1, 4) If I was given 3 points, I know how to do this. Simply find AB x AC and plug one ...
35 views

### Quick question regarding wording of a homework question

Find the equation for the plane passing through the heads of the three given vectors (2, 2, 0) (−1, 2, 1) (1, 1, 4) Is this just another way of asking what is the plane passing through these ...
47 views

34 views

### For all unit vectors $\mathbf u$ and a positive definite $\mathbf C$, what surface do vectors $\mathbf u \mathbf u^\top \mathbf C \mathbf u$ form?

Let $\mathbf C$ be a positive-definite $k\times k$ matrix. For all vectors $\mathbf u\in \mathbb R^k$ of length $\|\mathbf u\|=1$, consider vectors $\mathbf {uu}^\top\mathbf{Cu}$; they form a surface ...
67 views

### Convexity criterion for piecewise regular planar curves

A theorem of classical differential geometry states, that a simple, closed and regular $\mathcal{C}^2$-curve $\gamma:[0;1]\to\mathbb{R}^2$ is convex, iff its signed curvature doesn't change sign. ...
132 views

### Divergence in Definition of Laplace-Beltrami Operator

I am trying to derive an explicit formula for Laplace-Beltrami operator in global Cartesian coordinates for a special case of plane curve. I have found this article, and I would like to match their ...
26 views

### Finite open covers of a complex $C^{(1)}$ curve.

Consider a complex curve $\gamma \subset \mathbb{C}$, parametrized by $\alpha: [a,b]\to \mathbb{C}$, with $\alpha \in C^{(1)}$. Further, consider an finite open cover $\Phi$ of $\gamma=\alpha([a,b])$. ...
### Rotation of a Line Intersecting the Curve $y = x – \log(x)$ as $x \rightarrow \infty$.
Let a straight line ("line 1") in the $xy$-plane have one end fixed at the origin $(0,0)$, and the other at a variable point $(x, x – \log(x))$ on the curve $y = x – \log(x)$. The domain of $x$ is the ...