# Tagged Questions

An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities ...

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### Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
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### Divisor question on the normal projective curve

Let $X$ a normal projective curve over an infinite field $k$, let $x_1,\dots,x_n$ be pairwise distinct closed points in $X$ and let $n_1,\dots,n_r\in\mathbb Z$. Let $$D=\sum_in_ix_i$$ (considered ...
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### When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
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### integral point on conics

Suppose we have a conic $ax^2 + bxy + cy^2 + dx + ey + f = 0$ where $a,b,c,d,e,f \in \mathbb{Q}$. Is there a way of computing the integer points on this curve. Since it is affine an not projective we ...
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### Mobius transformation of Algebraic curve

I am working on the uniformization of algebraic curve problem. Currently, my adviser gave me a question about build a Mobius transformation between algebraic curves, and then lift it to the Rimeann ...
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### Finding the area enclosed by $px^4+qxy+ry^2+sy+t=0$

When $px^4+qxy+ry^2+sy+t=0\ (p,q,r,s,t\in\mathbb R)$ represents a simple closed curve on the $xy$ plane, can we represent the area enclosed by this curve by $p,q,r,s,t$? If yes, then how? Example 1 ...
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### Merge two or more cubic Bézier curves for optimization

I am looking for an algorithm which can merge several cubic Bezier curves. For instance, I have a lot of cubic Bezier that are joined to form a poly-Bezier curve. The idea is to merge dynamically some ...
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### is it possible to express the moduli of ppav's using torelli loci?

This is a probably vague question from an outsider: It is well known that it is possible to embed "the" moduli space of curves $M_g$ with fixed genus g into the moduli space of principally polarized ...
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### Parabola tangent to four lines

Suppose that in the affine plane R^2 four lines are given, with the property that no two are parallel and no three are concurrent. Show that there exists a unique parabola tangent to each of the four ...
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### Why is it called “elliptic” curve?

One of my favourite and most studied algebraic curve is the elliptic curve. But something that I have never asked myself is: Why do they call this nonsingular cubic curve an "elliptic" curve? ...
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### Why a cubic plane curve meets a line three times?

Can someone explain to me why a cubic curve in a projective plane always meets a line three times?
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### How can I find a tranformation matrix/Mathematical relation between two 5th degree polynomial curves in space?

I have the equation of two 5th degree polynomials which they don't intersect with each other. Each curve is made of 100 points and these two curves look similar but there are small differences. I am ...
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### How can I find a tranformation matrix/Mathematical relation between two 5th degree polynomial curves in space?

I have the equation of two 5th degree polynomials which they don`t intersect with each other .Each curve is made of 100 points and these two curves looks similar but there are small differences .I am ...
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### Simple Branched covering over sphere.

A simple branched covering is a branched covering with branching points of degree at most 2, in some context, it is also required to have at most one branching point in each fiber. My question is ...
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### Resample Bézier Curve with curvature and number of points constraints

I have an algorithm that implements an uniform resample process throughout a Bézier curve. This is done using a chord parametrization process. However, the results achieved do not accomplish my needs....
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### Find the arc length of a curve. Problem integrating

The question is find the arc length of the parabola $y^2 = 4ax$ cut by the line $3y = 8x$ I applied this formula $\int(1+ (dx╱dy)^2) dy$. However by substituting the value of $dx/dy I$ obtain an ...
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### Holomorphic Differentials on a non-singular curve.

So I've been working on this for an exam I have coming up and I'm not sure I really understand. If I have a curve defined by some homogenous polynomial P, I can show that the canonical divisor class ...