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.

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Differentiable, Parametrized Curve for Trace $y = |x|$

my professor gave practice problems for an upcoming midterm, and one is to find a differentiable parametrized curve with trace $y = |x|$ with $-1 \leq x \leq 1$. My thinking is that I should find a ...
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Are all curves with equation of the form $(\xi x +n) \cdot x = \text{const}$ circles?

Let $x(t)=(x_1(t),x_2(t))$ with $t\in [a,b]$ be a smooth curve in $\mathbb{R}^2$ and $\xi \in \mathbb{R}$ such that $$(\xi x +n) \cdot x = \text{const}$$ Here $n$ is the unit normal to the curve. Is ...
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Prove that for any piecewise smooth curve it is possible to find the parametrisation

Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie. length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal ...
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Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $\gamma (t) = (sin(t)+2,0,t)$ on XZ plane for $t \in R$, ...
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Can a curve cross its asymptote infinitely many times?

Can a curve cross infinitely many times its asymptote? If so, is there a special name for this behaviour?
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Can a surface of revolution be built from a self-intersected curve?

I'm reading "Differential Geometry of Curves And Surfaces" of Manfredo Do Carmo. There's a point in his book about Surfaces of Revolution which confuses me a lot. Here is the part: The part ...
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Problem about curves. A particle is running along circumference $x^2+y^2=25$

I'm considering a problem about curves. A particle is running along circumference $$x^2+y^2=25$$ with a costant modulus speed compliting a turn in 2 second. I need to determinate the acceleration in ...
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What's so special about involute curves??

An involute curve (specifically, an involute of a circle) is very commonly used to define the shape of the teeth on a gear. Apparently this idea goes back to Euler. Why is this? What special ...
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Plane curves admitting several ways to seal it up by the disc

Let $\gamma:S^1\to\mathbb R^2$ be an immersed closed curve with only isolated transversal self-intersections. Say that $\gamma$ is sealed by the disc if there is an immersion $f:D^2\to\mathbb R^2$ ...
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Parametric problem with circumference and tangents

Given the circumference $(x-3)^2+(y-2)^2=13$ find $k$ where $k$ is a coefficient in the parametric equation $(k+1)x+8ky-6k+2=0$ of the lines passing through the points $A(0;4)$, $B(6;4)$, $C(1;-1)$. ...
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Calculating double integral $\iint_{D}xy\,{\rm d}x\,{\rm d}y$ where $D$ is the plane limited by lines $y+x=1$, $y=0$, $x=0$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $a$ and graded ...
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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 ...
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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 $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
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Are all rationally parametrized plane curves algebraic? How does one find their degree?

Suppose a plane curve is given parametrically by $x=p(t),y=q(t)$, where $p,q$ are rational functions. I originally assumed that this means that the parametrized curve is algebraic, i.e. that it is the ...
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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 ...
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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 - ...
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Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
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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 ...
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Why is this the equation of the tangent plane?

I want to find the equation of the tangent plane of the surface patch $\sigma (r, \theta)=(r\cosh \theta , r\sinh \theta , r^2)$ at the point $(1,0,1)$. I have done the following: The point ...
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Problem with $arg(\gamma (t))$

I am see my notes about curves on complex spaces and I do not understand why it is so... I need help. I do not understand what way take to do it, I need someone explain slowly please, thanks
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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 ...
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Is this the correct way to compute the blow up of a curve

I'm trying to calculate the blowup of the curve $y^5=z^2-3z^3+2z^4$ at $(0,0)$ We have the relation $Ay=Bz$, now I split it into two charts: The first chart$(y,a=A/B)$: ...
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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?
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Angle between tangent vector and point of a cardiod.

Consider the cardioid $\rho=2a\left( 1 - \cos \phi \right)$. Show that the angle between the tangent vector and an arbitrary point (different from the origin) of the curve is half the polar angle. ...
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When is $t \mapsto \gamma (t) = \big( (1 + a \cos t) \cos t, (1 + a \cos t) \sin t \big)$ simple and closed?

Show that $\gamma (t) = \big( (1 + a \cos t) \cos t, (1 + a \cos t) \sin t \big), t \in [0, 2\pi]$, where $a$ is a constant, is a simple closed curve if $|a| < 1$ , but that if $|a| > 1$ its ...
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Image of a circle under conformal map $1/z$

The image of a circle under conformal map $1/z$ should be a circle, but how to prove it (or how to find the relationship between the two circles)? $z = x + iy = d + a\exp(i\theta)$, where $a$ is the ...
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Finding a parametric form for the locus of points for a vanishing determinant

I need to find the locus of points in the real $(x, y)$ plane, in parametric form, satisfied by the equation \det\begin{pmatrix} 0 & 1 & 1 & 1 & 1\\ 1 & 0 & ...
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