1. Homogeneous Coordinates
这个课件就是史. 我也不确定写的对不对$\left(\right.$
1.1. Real Projective Space
In mathematics, real projective space, denoted ${\displaystyle \mathbb {RP} ^{n}}$ or ${\displaystyle \mathbb {P} _{n}(\mathbb {R} )}$, is the topological space of lines passing through the origin $0$ in the real space ${\displaystyle \mathbb {R} ^{n+1}.}$ It is a compact, smooth manifold of dimension $n$, and is a special case ${\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})}$ of a Grassmannian space.
$\textbf{Examples}$ Low-dimensional real projective spaces are:
- $\mathbb{RP^1}$ is called the real projective line, which is topologically equivalent to a circle.
- $\mathbb{RP^2}$ is called the real projective plane. This space cannot be embedded in $\mathbb R^3$. It can however be embedded in $\mathbb R^4$ and can be immersed in $\mathbb R^3$. The questions of embeddability and immersibility for projective n-space have been well-studied.
We only study basic properties of $\mathbb{RP}^2$ in this article.
A point $P\in \mathbb{RP^2}$ in the plane is represented by a triple $(x, y, z)$, where $x, y, z$ are not all zero. Two triples $(x, y, z)$ and $(x', y', z')$ represent the same point if and only if there exists a nonzero scalar $\lambda$ such that $x = \lambda x', y = \lambda y', z = \lambda z'$. This is called the homogeneous coordinates of a point.
2. Projective Plane
Suppose the observer’s viewpoint is situated at the origin, and the
space onto this canvas can be expressed as the following transformation:
$\textbf{Example}$ The line $(-1, 0, t)$ for $t\in \mathbb{R}$ in Cartesian coordinates is projected to the line $(-1/t, 0, 1)$ in homogeneous coordinates, and will converge to the point $(0, 0, 1)$ as $t\to \infty$. Also the line $(1, 0, t)$ for $t\in \mathbb{R}$ is projected to the line $(1/t, 0, 1)$ in homogeneous coordinates, and will also converge to the point $(0, 0, 1)$ as $t\to \infty$, illustrating the concept of lines intersecting at a point at infinity.
$\textbf{Definition}$: We define the affine plane (仿射平面)
It’s named this because it remains invariant under the action of scaling by a nonzero scalar. The projective plane $\mathbb{RP^2}$ is defined as the union of the affine plane $\mathbb A^2$ and the set of all points at infinity, which is denoted by $\mathbb{RP^1}$.
$\textbf{Theorem}$: $\mathbb{RP^2}=\mathbb{A}^2\cup \mathbb{RP^1}$
Proof: The set of all equivalence classes $(x: y: z)$ in $\mathbb R^3$ can be divided into two categories
- The set of all equivalence classes $(x: y: 0)$, which is isomorphic to $\mathbb{RP^1}$.
- The set of all equivalence classes $(x: y: 1)$ with $z\neq 0$, $x\leftarrow \frac x z$, $y\leftarrow \frac y z$, which is isomorphic to $\mathbb{A}^2$.
Or more specifically, $\mathbb{RP^2}=\mathbb{A}^2\sqcup \mathbb{RP^1}$
3. General Cartesian Equation of a Conic
It’s known that he general cartesian form of a conic can be written in matirx form as
where $A, B, C, D, E, F$ are real numbers, and the matrix $\left(\begin{array}{cc}A & \frac{B}{2} \\ \frac{B}{2} & C\end{array}\right)$ is symmetric and positive definite.
Or we can write it as
where the matrix $\left(\begin{array}{ccc}A & B/2 & D/2\\ B/2 & C & E/2 \\ D/2 & E/2 & F\end{array}\right)$ is symmetric and positive definite.
If we turn to the homogeneous coordinates, the equation becomes
This is very natural because the equation is invariant under scaling by a nonzero scalar~