Table of Contents
1. Line
1.1. Cartesian Coordinates
1.1.1. Standard form
- \(y=mx+n\)
1.1.2. General form
- \(ax+by+c=0\)
1.1.3. Intercept form
- \[ \frac{x}{a}+\frac{y}{b}=1 \]
1.1.4. Ratio form
- \[ \frac{y-y_0}{x-x_0}=k \]
1.1.5. Point-Gradient form
- \((y-y_0)=m(x-x_0)\)
1.1.6. Two-Point form
- \[ (y-y_1)=\frac{y_2-y_1}{x_2-x_1}(x-x_1) \]
1.1.7. Point-Displacement Form
- \[ t=\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}=\dots \]
1.1.8. Parameterized Form
\[
\begin{align*} x&=at+x_0\\ y&=bt+y_0\\ z&=ct+z_0\\ &\dots \end{align*}\]
1.2. Vector
1.2.1. Vector Parameterized form
- \(\mathbf{x}=\mathbf{x}_0+k\mathbf{d}\)
- \(\mathbf{d}\) consists of the denominators from
((652778ff-870f-4b88-89de-08a76ccccd86)).
- \[ \mathbf{d}=\begin{bmatrix}a\\b\\c\\\vdots\end{bmatrix} \]
1.2.2. Affine Form
- Or Convex combination form, Barycentric form
- \(\mathbf{x}=t\mathbf{x_1}+(1-t)\mathbf{x_2}\)
1.2.3. Circle Secant Form
- Line that passes through two points on a circle, such that two
tangent lines at the points passes through some point \(\bf a\):
- \((\mathbf{a}-\mathbf{x_0})\cdot(\mathbf{x}-\mathbf{x_0})=r^2\)
- From the ((652781e2-c54f-43c7-8265-22d95d32c26c)) of the equation of circle.
- Although it started from a notion of tangent lines, but it also works for the points inside of a circle.
- Related to
1.2.4. Circle Tangent Form
- Tangent line of a circle
- \((\mathbf{x}_1-\mathbf{x}_0)\cdot(\mathbf{x}-\mathbf{x_0})=r^2\) where \(\mathbf{x}_1\) is a point on the circle.
1.3. Plücker Coordinates
- Six homogeneous coordinates that corresponds to a line in projective 3-space \(\mathbb{P}^3\).
- It consists of a displacement \(\mathbf{d}\) and a moment
\(\mathbf{m}\): \((d_1: d_2: d_3: m_1: m_2: m_3)\).
- For two points \(\mathbf{x}, \mathbf{y}\) on the line,
\(\mathbf{d} = \mathbf{y} - \mathbf{x}\) and
\(\mathbf{m} = \mathbf{x}\times \mathbf{y}\).
- The moment is perpendicular to the plane that passes through the line and the origin.
Alternatively, since the equation of line can be described as the intersection of two planes: \[
\begin{cases} \mathbf{a\cdot x} + a = 0\\ \mathbf{b\cdot x} + b = 0 \end{cases}\]
- \(\mathbf{d} = \mathbf{a\times b}\) and \(\mathbf{m} = a\mathbf{b} - b\mathbf{a}\).
- For two points \(\mathbf{x}, \mathbf{y}\) on the line,
\(\mathbf{d} = \mathbf{y} - \mathbf{x}\) and
\(\mathbf{m} = \mathbf{x}\times \mathbf{y}\).
2. Plane
2.1. Orthogonal Form
- \(\mathbf{n}\cdot(\mathbf{x}-\mathbf{x}_0)=0\)
3. Algebraic Variety
- Classically, a set of solutions of a system of polynomial equations.
3.1. Algebraic Set
- The zero-locus in affine \(n\)-space \(\mathbb{A}^n\) of the set \(S\) of polynomials in \(K[x_1, \dots, x_n]\): \[ V(S) := \{x\in \mathbb{A}^n \mid \forall f\in S, f(x) = 0\}. \]
3.1.1. Irreducibility
- A nonempty affine algebraic set \(V(S)\) is called irreducible if it cannot be written as the union of two proper algebraic subsets.
3.2. Defintion
- Set of roots of the ideals, in an algebraically closed field.
3.2.1. Affine Variety
- Irreducible affine algebraic set \(V(S)\) for any set of polynomials \(S\)
3.2.2. Abstract Definition
- integral, separated scheme of finite type over an algebraically closed field