0
votes
0answers
28 views

How to find the tangent vector to a curve at a special point

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. The line segments $AC$ and $BC$ make equal angle $\alpha$ with the horizontal plane through $C$. The angle ...
1
vote
1answer
66 views

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 ...
1
vote
1answer
53 views

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 ...
1
vote
1answer
67 views

An question on effective divisor (Clifford 'S theorem)

For an effective divisor $D\ge 0$ on a curve $Y$, define $$\lvert D\rvert =\{ D' \in \mathrm{Div}(Y) \mid D'\ge 0 \;\text{ and }\; D' \sim D \}$$ where $D\sim D$ means $\exists$ a rational ...
0
votes
1answer
37 views

Defined matrix in Catmull Spline Curve

I am trying to use Catmull spline curve in my program , I am trying to understand it but why we only use below given Matrix , because the examples I saw I only found the below one In Catmull spline ...
1
vote
1answer
54 views

multiplicity of intersection of projective curves

This example is taken from Silverman and Tate's Rational Points on Elliptic Curves. Suppose we have the line $C_1: x+y=2$ and the circle $C_2: x^2 + y^2 = 2$. We see that these intersect at the ...
0
votes
1answer
58 views

Bézier curve limits

Can be any curve of any shape (without sharp edges) described by Bézier curve with unlimited (but finite) number of control points? The answer to the question above would probably be no, because I ...
1
vote
0answers
28 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
6
votes
1answer
153 views

Exercise 1.11 of Eisenbud

I'm doing the exercises from Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, and I don't understand part of one of them, ex. 1.11 a): Exercise 1.11 a: Over $\mathbb{C}$, ...
1
vote
0answers
59 views

How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
5
votes
1answer
71 views

Given a non-singular curve $C$, show that two divisors are algebraically equivalent iff they have the same degree

I wish to show that given a non-singular curve $C$, two divisors are algebraically equivalent if and only if they have the same degree. I'm rather stuck on how to approach such a problem. I'm ...
1
vote
0answers
51 views

Projections of a curve in $\mathbb{P^3}$

Given a point $P_0$ and an hyperplane $H$ in $\mathbb{P^3(C)}$, it is possible to define the projection of center $P_0$ on $H$: $$\pi:\mathbb{P^3(C)}\setminus\{P_0\}\rightarrow H,\quad ...
-1
votes
1answer
203 views

Comparing 2 non-linear curves

I have 2 non-linear curves having (x,y)values. The x values are varying from 0 to 127 in both the curves and y values are of different magnitude for 2 curves. How can I compare these 2 non-linear ...
7
votes
1answer
157 views

On the automorphisms of the Klein Quartic

I am trying to solve a problem from Miranda's book, Algebraic Curves and Riemann Surfaces. On page 84, problem K gives the Klein curve $X$ as a smooth projective plane curve defined by the equation ...
2
votes
0answers
75 views

Puiseux series and Resolution of Singularities

I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of ...
6
votes
2answers
147 views

A question on Newton's “theorem about ovals”

This is a question about a result from Newton's Principia. It says, roughly, that the if you intersect lines $ax + by + c$ with a smooth, closed, convex curve, then the area of the curve that the line ...
0
votes
2answers
172 views

some notions on algebraic curve

1) I want to learn about algebraic curves and i'm confused, please correct me if i'm wrong : when we say an Affine algebraic curve over the field $F$ : here affine to distinguish it from projective ...
0
votes
3answers
909 views

Intersection of Cubic curves

This is the question which i am attempting to solve, and it seems to difficult to get rid of the exponents. Show that a the two cubic curves $Y^3 = X^2 + X^3$ and $X^3 = Y^2 + Y^3$ intersect in ...