My own revision on the module Geometry, mainly from example sheets and past papers
This repository will mainly focus on two parts, from support class sheets and past papers. For the support class I will create my own pdf, so no license issues should be addressed.
- MA243-Geometry-Revision
- Lecture notes
- Example sheets
- Week 1
- Week 2
- Week 3
- Week 4
- Week 5
- Week 6
- Week 7
- Equivalence relation
- Projective space
- k-dimensional projective linear subspace
- Definition of point, line and plane in projective geometry
- Dimension formula for projective geometry
- Projective general linear group
- Projective transformation
- Projective frame of reference
- Standard frame of reference
- Bijection between projective transformation and projective frames of reference
- Week 8
First thing first, feel free to download the 2022 version of lecture nots.
In week 1's example sheets, there are some new definitions and old linear algebra knowledge to look at.
- A metric or distance,
$d$ on a set$X$ is a function$$d:X\times X\to[0,\infty)$$ such that it is
(1) Non-degenerate:
- If
$\mathbf{x},\mathbf{y},\mathbf{z}$ are distinct collinear points on a line$L\subset\mathbb{R}^{m}$ , then for some$\lambda\in\mathbb{R}$ we have$$\mathbf{z} = \lambda_{1}\mathbf{x}+\lambda_{2}\mathbf{y},$$ where$\lambda_{1}+\lambda_{2} = 1$ and$\lambda_{2} = \lambda$ .
- If
$(X_{1},d_{1})$ and$(X_{2},d_{2})$ are metric spaces, then a distance preserving map between$(X_{1},d_{1})$ and$(X_{2},d_{2})$ is a map$f:X_{1}\to X_{2}$ such that for any points$P,Q$ in$X_{1}$ , we have$$d_{2}(f(P),f(Q)) = d_{1}(P,Q).$$ - An isometry is a bijective distance preserving map.
- If
$f$ is an isometry, then$(X_{1},d_{1})$ and$(X_{2},d_{2})$ are said to be isometric. - A Euclidean space is a metric space which is isometric to
$\mathbb{R}^{n}$ , with the Euclidean metric, for some integer$n$ . We use the notation$\mathbb{E}^{n}$ to denote the metric space$\mathbb{R}^{n}$ together with the Euclidean metric. If not specified otherwise,$\mathbb{R}^{n}$ is used to mean$\mathbb{E}^{n}$ .
- A map
$T:\mathbb{R}^{n}\to\mathbb{R}^{k}$ is affine as it is of the form$T(\mathbf{x}) = L(\mathbf{x}) + \mathbf{b}$ for all$\mathbf{x}\in\mathbb{R}^{n}$ for some linear map$L:\mathbb{R}^{n}\to\mathbb{R}^{k}$ and$\mathbf{b}\in\mathbb{R}^{k}$ . Note that$\mathbf{b} = T(\mathbf{0}).$
- A reflection matrix has eigenvalues
$\pm 1$ , while a rotational matrix with an anti-clockwise rotation has eigenvalues$e^{\pm i\theta}$ . - A reflection matrix has determinant -1 while a rotation matrix has determinant 1.
- An Euclidian motion is a bijective map
$T:\mathbb{E}^{n}\to\mathbb{E}^{n}$ that is an isometry.
- We can assume that the glide
$f$ is given by
where
In week 2's example sheets, note that there are some theorems to be revised.
- Vectors
$\mathbf{u},\mathbf{v}$ are orthogonal to each other if their inner product is 0. - A basis of
$\mathbb{R}^{n}$ is an orthogonal basis if it consists of normalised vectors which are mutually orthogonal. - An orthogonal matrix is
$A^{T} = A^{-1}$ .
- If
$V$ is a vector space, with a subspace$W$ , then the orthogonal complement of$W$ in$V$ is the vector subspace$$W^{\perp} = \set{\mathbf{v}\in V:\langle\mathbf{v},\mathbf{w}\rangle = 0,\forall \mathbf{w}\in W}.$$
- If a matrix has no real eigenvalues, then it is not diagonalisable over the real numbers.
- Any real polynomial of odd degree has a real root.
- Let
$L:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a linear map with matrix$A$ with respect to the standard basis. Then the following are equivalent: -
$L$ is an isometry. -
$\left|L(\mathbf{x})\right| = \left|\mathbf{x}\right|$ for all$\mathbf{x}\in\mathbb{R}^{n}$ , i.e.$L$ is norm preseving. -
$\langle L(\mathbf{x}), L(\mathbf{y})\rangle = \langle \mathbf{x},\mathbf{y}\rangle.$ ,$\forall\mathbf{x},\mathbf{y}\in\mathbb{R}^{n}$ , i.e.$L$ preserves the inner product. - The matrix
$A$ is orthogonal, i.e.$A^{T}A = I$ .
- Let
$L:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a linear isometry and$A$ an orthogonal matrix such that$L(\mathbf{x}) = A\mathbf{x}$ . Then there is an orthonormal basis of$\mathbb{R}^{n}$ with respect to which$L$ has the matrix of the form
where
-
$A$ is normal if and only if it is unitarily diagonalisable. i.e.$A$ is normal if and only if there exists a unitary matrix$U$ such that$$A = UDU^* ,$$ where$D$ is a digaonal matrix, and$U$ is a unitary matrix, meaning its conjugate transpose$U*$ is also its inverse, that is, if$$U^* U = UU^* = UU^{-1} = I.$$
- Let
$T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a Euclidian isometry. Then the image$T(C)$ of a circle$C$ of centre$x\in\mathbb{R}^{2}$ and radius$r > 0$ is a circle of centre$T(x)$ and radius$r$ . (This is because let $y\in C$m then$d(y,x) = r$ . Since$T$ is an isometry,$d(T(x),T(y)) = d(y,x) = r$ , so$T(y)\in C'$ of radius$r$ centred at$T(x)$ . Conversely, if$z\in C'$ then since$T^{-1}$ is an isometry,$d(T^{-1}(z), T^{-1}(x)) = d(z,x) = r$ , so$T^{-1}(z) \in C$ of radius$r$ centred at$T^{-1}(x)$ . Thus,$z = TT^{-1}(z)$ is in the image of$C$ .) - Let
$n\geq 1$ and$L:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a linear Euclidian isometry with a real eigenvalue$\lambda$ , then$\lambda = \pm 1$ . - For every
$n\geq 2$ , every Euclidian line$L$ in$\mathbb{R}^{n}$ and point$P\in\mathbb{R}^{n}$ not belonging to$L$ , there is a line$L'$ passing through$P$ such that$L\cap L' = \emptyset.$
In week 3's example sheet, we focus mainly on groups and hyperplanes.
- A hyperplane of
$\mathbb{R}^{n}$ is an affine subspace of dimension$n-1$ , which has the form$$\Pi = V+\mathbf{b} = \set{\mathbf{b}+\mathbf{v}:\mathbf{v}\in V} = (\mathbb{R}\mathbf{v}_ n)^{\perp} + \mathbf{b} = \set{\mathbf{v}\in\mathbb{R}^{n}:\langle\mathbf{v},\mathbf{v}_ n\rangle = \beta}.$$
- The reflection
$\rho_{\Pi}$ in the hyperplane$\Pi$ exists and is unique, and is given by$$\rho_{\Pi}:\mathbf{P}\to\mathbf{P}-2\langle\mathbf{P}-\mathbf{b},\mathbf{v}\rangle\mathbf{v},$$ where$\Pi = \mathbf{b}+(\mathbb{R}\mathbf{v})^{\perp}$ and$\left|\mathbf{v}\right| = 1$ . - If
$R$ is a reflection and$T$ is any isometry, then$T^{-1}\circ R\circ T$ is also a reflection. - If
$\rho_{\Pi}$ is a reflection in the plane$\Pi$ , then the conjugation of$\rho_{\Pi}$ by the reflection$T(\mathbf{x}) = \mathbf{x}+\mathbf{c}$ , that is, the map$T^{-1}\circ\rho_{\Pi}\circ T$ is a reflection in$\Pi - \mathbf{c}$ . - A reflection
$R$ has order 2, that is,$$R\circ R = \text{id}.$$
- A homomorphism between two groups
$G$ and$H$ is a map$$\phi:G\to H$$ such that$\phi(gh^{-1}) = \phi(g)\phi(h)^{-1}$ for all$g,h\in G$ . i.e.$\phi$ preserves multiplication, inverse, identity. - An Isomorphism is a group homomorphism which is a bijection.
- The group of automorphisms of a group
$H$ is the group$$Aut(H) = \set{f:H\to H:f\ \text{is}\ \text{group}\ \text{isomorphism}},$$ with group law composition of functions.
In week 4's example sheet, we enter the field of spherical geometry.
- The n-dimensional sphere of radius
$r\geq 0$ is defined by$$S_{r}^{n} = \set{\mathbf{x}\in\mathbb{R}^{n+1}:\left|\mathbf{x}\right| = r},$$ where$S^{n} = S_{1}^{n},$ which has Euclidean norm 1. - A spherical line or great circle is the intersection of
$S_{r}^{n}$ with a 2-dimensional vector subspace of$\mathbb{R}^{n+1}$ , i.e.$$S^{2}\cap\Pi,$$ where$\Pi$ is some plane in$\mathbb{R}^{3}$ containing 0.
- If
$\mathbf{P},\mathbf{Q}\in S_{r}^{n}$ , then they are antipodal if$\mathbf{Q} = -\mathbf{P}$ . - If
$\mathbf{P}$ and$\mathbf{Q}$ in$S_{r}^{n}$ are not antipodal, then there is a unique great circle containing both of them.
- The spherical distance between two points
$\mathbf{P},\mathbf{Q}\in S_{r}^{n}$ is the length of the shortest arc of a great circle joining them. We will show that this distance is a metric, called the spherical metric, which we can also define by$$d_{S_{r}^{n}}(\mathbf{P},\mathbf{Q}) = r\arccos\left(\frac{\langle\mathbf{P},\mathbf{Q}\rangle}{r^{2}}\right).$$
- Let
$\alpha,\beta,\gamma$ be the side lengths of a spherical triangle with vertices$\mathbf{P},\mathbf{Q},\mathbf{R}\in S^{2}$ on the unit sphere,$$\alpha = d(\mathbf{Q},\mathbf{R}),\quad\beta = d(\mathbf{P},\mathbf{Q}),\quad\gamma = d(\mathbf{P},\mathbf{Q}),$$ where$d = d_{S^{2}}$ is the spherical metric. Let$a$ be the spherical angle between arcs$\mathbf{P}\mathbf{Q}$ and$\mathbf{P}\mathbf{R}$ . Then$$\cos\alpha = \cos\beta\cdot\cos\gamma + \sin\beta\cdot\sin\gamma\cdot\cos a.$$
- An isometry
$T:S^{n}\to S^{N}$ preserves antipodal points. - An isometry
$T:S^{n}\to S^{N}$ preserves great circles. - A bijective map
$f:\mathbb{R}^{k}\to\mathbb{R}^{k}$ , which preserves the standard inner product on$\mathbb{R}^{k}$ is a linear isometry of$\mathbb{R}^{k}$ . - If
$L:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ is a linear isometry and since it is bijective, then$$L(A\cap B) = L(A)\cap L(B)$$ for general subsets$A,B$ .
- The spherical triangle
$\triangle$ with angles$\alpha,\beta,\gamma$ is$$\text{area}(\triangle) = \alpha+\beta+\gamma - \pi.$$
- If
$\mathcal{C}$ and$\mathcal{D}$ are two distinct great circles on$S^{2}$ , then$\mathcal{C}\cap\mathcal{D} = \set{-P,P}$ for some$P\in S^{2}$ . (This is because a great circle is the intersection of$S^{2}$ with a 2-dimensional vector subspace of$\mathbb{R}^{3}$ . If$C = S^{2}\cap\Pi$ and$c' = S^{2}\cap\Pi'$ are two distinct great circles, the planes$\Pi$ and$\Pi'$ are also distinct, and their intersection is a line$L$ of$\mathbb{R}^{3}$ passing by 0. Thus the intersection would be$C\cap C' = S^{2}\cap\Pi\cap\Pi' = S^{2}\cap L$ . If$v$ is a unit vector on$L$ , this intersectio nconsists of$v$ and$-v$ , which are the two unit vectors on$L$ .)
In week 5's example sheet, we mainly focus on spherical geometry and hyperplane and Lorentz transformation.
- The hyperbolic metric on
$\mathcal{H}^{n}$ is defined as$$\mathcal{H}^{n} = \set{\mathbf{x}\in\mathbb{R}^{n+1}:\left|\mathbf{x}\right|_ {L} = i, x_{1} > 0}.$$
- We note the difference between these two:
$$d_{S^{n}}(\mathbf{x},\mathbf{y}) = \cos^{-1}(\mathbf{x},\mathbf{y}) = \left(\frac{\langle\mathbf{x},\mathbf{y}\rangle}{|\mathbf{x}| |\mathbf{y}|}\right)\quad d_{\mathcal{H}^{n}}(\mathbf{x},\mathbf{y}) = \cosh^{-1}(\mathbf{x},\mathbf{y}) = \left(\frac{\langle\mathbf{x},\mathbf{y}\rangle_{L}}{|\mathbf{x}|_L|\mathbf{y}|_L}\right).$$
There are several terminology including time,space and light like vectors:
- A point
$\mathbf{x}\in\mathbb{R}^{n}$ is space-like if$\left|\mathbf{x}\right|_{L}\in (0,\infty)$ , i.e. if$\langle\mathbf{x},\mathbf{x}\rangle _{L}>0.$ - A point
$\mathbf{x}\in\mathbb{R}^{n}$ is light-like if$\left|\mathbf{x}\right|_{L} = 0$ , i.e. if$\langle\mathbf{x},\mathbf{x}\rangle _{L}=0.$ - A point
$\mathbf{x}\in\mathbb{R}^{n}$ is time-like if$\left|\mathbf{x}\right|_{L}\in i(0,\infty)$ , i.e. if$\langle\mathbf{x},\mathbf{x}\rangle _{L}<0.$ - A point
$\mathbf{x}\in\mathbb{R}^{n}$ is positive if$x_{1} > 0.$ - A point
$\mathbf{x}\in\mathbb{R}^{n}$ is negative if$x_{1} < 0.$ - A time-like or space-like vector
$\mathbf{x}\in\mathbb{R}^{n}$ is positive if$x_{1} > 0$ and negative if$x_{1} < 0.$
- A map
$T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a Lorentz transformation if it preserves the Lorentz inner product:$$\langle T(\mathbf{x}),T(\mathbf{y})\rangle_{L} = \langle\mathbf{x},\mathbf{y}\rangle_{L},\forall\mathbf{x},\mathbf{y}\in\mathbb{R}^{n}.$$ - We see that
$T$ is positive if$\mathbf{x}$ is positive time-like if and only if$T(\mathbf{x})$ is positive time-like. - A Lorentz transformation is bijective.
- An
$n\times n$ real matrix$A$ is Lorentz orthogonal if
- The Lorentz group is the group of Lorentz orthogonal
$(n+1)\times (n+1)$ matrices, denoted$$O(1,n) = \set{A\in M_{(n+1)\times(n+1)}|A^{T}JA = J}.$$ - The positive Lorentz group is the subgroup of
$O(1,n)$ which maps positive time like vectors bijectively to positive time like vectors. This is denoted$O^{+}(1,n)$ . -
$O(1,n)$ and$O^{+}(1,n)$ are groups. - A set of vectors
$\mathbf{v}_ 1,...,\mathbf{v}_ n\in\mathbb{R}^{n}$ is Lorentz orthogonal if$\langle\mathbf{v}_ {i},\mathbf{v}_ {j}\rangle_{L} = 0$ for$i\ne j$ . It is Lorentz orthonormal if - (1)
$\langle\mathbf{v}_ {i},\mathbf{v}_ {j}\rangle_{L} = 0,$ if$i\ne j$ . - (2)
$\langle\mathbf{v}_ {i},\mathbf{v}_ {j}\rangle_{L} = 1,$ if$2\leq i=j < n$ . - (3)
$\langle\mathbf{v}_ {i},\mathbf{v}_ {j}\rangle_{L} = -1,$ if$i=j=1$ . - If
$A\in O(1,n)$ , then$A^{T}\in O(1,n).$
- The canonical basis
$\mathbf{e}_ {1},...,\mathbf{e}_ {n}$ is Lorentz orthonormal. - If
$\mathbf{v}_ {1},...,\mathbf{v}_ {n}$ are Lorentz orthonormal, then they form a basis of$\mathbb{R}^{n}$ . -
$\mathbf{v}_ {1},...,\mathbf{v}_ {n}$ is a Lorentz orthonormal basis if and only if$\langle\mathbf{v}_ {i},\mathbf{v}_ {j}\rangle_{L} = J_{i,j}$ , the element in the ith row, jth column of$J$ . - A linear map
$T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ , with matrix$A$ is a Lorentz transformation if and only if it maps the standard basis to a Lorentz orthonormal basis. - Let
$A$ be a$n\times n$ matrix. THe following are equivalent: - (1) The map
$T(\mathbf{x}) = A\mathbf{x}$ is a Lorentz transformation. - (2)
$\left|T(\mathbf{x})\right|_ {L} = \left|\mathbf{x}\right|_ {L}$ for all$\mathbf{x}\in\mathbb{R}^{n}$ . - (3)
$A$ is Lorentz orthogonal. - A map
$T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a Lorentz transformation if and only if - (1) The image of the canonical basis
$T(\mathbf{e}_ {1}),...,T(\mathbf{e}_ {n})$ is Lorentz orthonormal. - (2)
$T$ is linear. - There is a basis of
$\mathbb{R}^{n+1}$ consisting of time-like vectors, which may be taken to be elements of$\mathcal{H}^{n}$ .
- A 2-dimensional sub-vector space
$V$ of$\mathbb{R}^{n+1}$ is called a Lorentz plane if it contains a time-like vector. A hyperbolic line is the intersection of$\mathcal{H}^{n}$ with any Lorentz plane, i,e,$$L = \pi\cap\mathcal{H}^{2}.$$
Week 6's example sheet is a recap of Lorentz geometry and isomorphism. Therefore all the lemmas and theorems are given above, there is only one to note.
- There is a paraterisation
$f:\mathbb{R}^{2}\to\mathcal{H}^{2}$ where$$f:(t,\theta)\to(\cosh(t),\cos(\theta)\sinh(t),\sin(\theta)\sinh(t)),\quad t\in[0,\infty),\theta\in[0,2\pi),$$ where$\mathbb{R}^{2}$ is given with polar coordiantes, and$\mathcal{H}^{2}$ with the Cartesian coordinates of$\mathbb{R}^{3}$ .
- The area of a hyperbolic triangle
$\triangle$ with angles$\alpha,\beta,\gamma$ is$$\text{area}(\triangle) = \pi-(\alpha+\beta+\gamma).$$
- For distinct points
$\mathbf{x},\mathbf{y},\mathbf{z}$ in$\mathcal{H}^{n}$ , let$$\alpha = d_{\mathcal{H}^{n}}(\mathbf{z},\mathbf{y}),\quad\beta = d_{\mathcal{H}^{n}}(\mathbf{x},\mathbf{z}),\quad\gamma = d_{\mathcal{H}^{n}}(\mathbf{x},\mathbf{y}),\quad a = \text{hyperbolic angle at x},$$ then$$\cosh\alpha = \cosh\beta\cdot\cosh\gamma - \sinh\beta\cdot\sinh\gamma\cdot\cos a.$$
- For every
$n\geq 2$ , every hyperbolic line$L$ in$\mathcal{H}^{n}$ and point$P\in\mathcal{H}^{n}$ not belonging to$L$ , there is a hyperbolic line$L'$ passing through$P$ such that$L\cap L' = \emptyset$ . (This is because let$L = \mathcal{H}^{n}\cap\Pi$ for some 2-dimensional subspace$\Pi$ of$\mathbb{R}^{n+1}$ . Since the dimension of$\mathbb{R}^{n+1}$ is at least 4, we can choose$v\in\mathbb{R}^{n+1}$ linearly independent from the elements of$\Pi$ and$P$ . Thus the span of$P$ and$v$ intersects$\Pi$ trivially, and therefore defines a hyperbolic line disjoint from$L$ . - Two distinct hyperbolic lines of
$\mathcal{H}^{2}$ intersect in at most one point. (This is because let$L = \Pi\cap\mathcal{H}^{2}$ and$L' = \Pi'\cap\mathcal{H}^{2}$ be two distinct hyperbolic lines. Then$L\cap L' = \Pi\cap\mathcal{H}^{2}\cap\Pi'\cap\mathcal{H}^{2} = (\Pi\cap\Pi')\cap\mathcal{H}^{2}$ , where$\Pi\cap\Pi'$ is the intersection of two 2-dimensional subspaces of$\mathbb{R}^{3}$ , which is a line through the origin. Let$v$ be a non-zero vector, and$\lambda v$ a point on the line it spans. Then$\left|\lambda v\right|_ {L} = |\lambda|\left|v\right|_ {v}$ and$|\lambda|\left|v\right|_ {L} = i$ has at most the real solutions$|\lambda| = \frac{i}{\left|v\right|_ {L}}$ , when$\left|v\right|_ {L}$ is pure imaginary. In this case there is exactly one of the two vectors$x = \pm(\frac{i}{\left|v\right|_ {L}}) v$ with$x_{1} > 0$ , which is the intersection point in$\mathcal{H}^{2}$ .)
Week 7's example sheet leads us to Projective Geometry.
- Define an equivalence relation on
$V^{*}$ by defining for$\mathbf{x},\mathbf{y}\in V$ ,$$\mathbf{x}\sim\mathbf{y}\Longleftrightarrow k\mathbf{x} = k\mathbf{y}.$$
- The projective space of
$V$ is the set of equivalence classes in$V^{*}$ under this relation:$$\mathbb{P}^{n}(k) = \mathbb{P}(V) = V^{ *}\setminus\sim.$$ - When
$k = \mathbb{R}$ , we write$\mathbb{P}^{n} = \mathbb{P}(\mathbb{R}^{n+1})$ , and call this$n$ -dimensional real projective space.
Alternative definition
- The projective space
$\mathbb{P}^{n}$ is the set of lines through 0 in$\mathbb{R}^{n+1}$ .
- If
$V$ is a vector space$V$ , with a$k+1$ dimensional vector subspace$W$ , then the k-dimensional projective linear subspace$\mathbb{P}(W)$ of$\mathbb{P}(V)$ is the image under$\pi$ of$W$ in$\mathbb{P}(V)$ , that is$$\mathbb{P}(W) = \set{\mathbf{x}\in\mathbb{P}(V):\mathbf{x}\subset W} = \pi(W).$$ - The dimension of
$\mathbb{P}(W)$ is one less than the dimension of$W$ .
- Let
$W$ be a subvector space of$V$ . Then$$\mathbb{P}(W)$$ is a -
point if
$\dim(W) = 1$ . -
line if
$\dim(W) = 2$ . -
plane if
$\dim(W) = 3$ .
- For projective linear subspaces
$E$ and$F$ of$\mathbb{P}(V)$ ,$$\dim(E\cap F) = \dim(E)+ \dim(F) - \dim\langle E,F\rangle,$$ with the convention$\dim(\emptyset) = -1$ and$\langle E,F\rangle$ is the linear subspace of$\mathbb{P}(V)$ spanned by$E$ and$F$ .
- The projective general linear group of a vector space
$V$ over a field$k$ is the group of all invertible linear maps from$T:V\to V$ up to scalar multiplication, that is, we consider$T_{1}\sim T_{2}$ if there is some non-zero element$\lambda\in k$ with$T_{2} = \lambda T_{1}$ . - We write PGL
$(n)$ = PGL$(\mathbb{R}^{n})$ . In this case, the elements of PGL$(n)$ are invertible matrices up to scalar multiplication, that is$$A\sim B\Longleftrightarrow A = \lambda B$$ for some$\lambda\in\mathbb{R}^{*}.$
- If
$A$ is an invertible$(n+1)\times(n+1)$ matrix, then the map$$T_{A}:\mathbb{P}^{n}\to\mathbb{P}^{n},\quad \left[\mathbf{v}\right]\to\left[A\mathbf{v}\right]$$ is called a projective transformation, or a projectivity or a projective linear map.
- A projective frame of reference for
$\mathbb{P}^{n}$ is an ordered set of$n+2$ points,$P_{0}, P_{1},..., P_{n+1}\in\mathbb{P}^{n}$ , any$n+1$ of which are linearly independent and so span$\mathbb{P}^{n}$ .
- The standard frame of reference for
$\mathbb{P}^{n}$ is given by$[\mathbf{e}_ {1}],...,[\mathbf{e}_ {n+1}]$ together with$[\mathbf{e}_ {1} + ...+\mathbf{e}_ {n+1}]$ , where$\mathbf{e}_{i}$ are the standard basis for$\mathbb{R}^{n+1}$ .
- There is a bijection between projective transformations of
$\mathbb{P}^{n}$ and projective frames of references of$\mathbb{P}^{n}$ , defined as follows:$$\phi: PGL(n+1)\to\set{\text{projective frames of reference}},\quad T\to\set{T([\mathbf{e}_ {1}]),...,T([\mathbf{e}_ {n+1}]), T([\mathbf{e}_ {1}+...+\mathbf{e}_ {n+1}])}.$$
This is the final example sheet as the remaining two focuses on assignments.
- A perspectivity
$f$ is a map between two distinct hyperplanes (linear subspaces of dimension$n-1$ ),$\Pi_{1}$ and$\Pi_{2}$ in$\mathbb{P}^{n}$ , given by projection from a point$O\notin\Pi_{1}\cup\Pi_{2}$ . That is , if$P\in\Pi_{1}$ , then$f(P)\in\Pi_{2}$ is the point such that$0,P,f(P)$ all lie on the same line.
- Two projectivities between lines coincide iff they coincide on a frame of reference, i.e. on three distinct points.
- Let
$P,Q,R,S$ be four distinct points on a line in$\mathbb{P}^{n}$ . Choose an appropriate basis such that$P = (1:0)$ and$Q = (0:1)$ . Since$\tilde{P}$ and$\tilde{Q}$ span the line that$R$ and$S$ lie on, and since$R,S\ne P$ , there are$\lambda,\mu\in\mathbb{R}$ such that we can write with respect to this basis:$$R = (1:\lambda), S = (1:\mu).$$ Then the cross ratio is$$\set{P,Q;R,S} = \frac{\lambda}{\mu}.$$ - Here,
$\tilde{P}$ is defined up to direction, not magnitude, and given by$$\tilde{P} = \text{any choice of non-zero}\mathbf{v}\in P.$$ - If
$P,Q,R,S$ are points in$\mathbb{P}^{1}$ , then they can be considered as lines in$\mathbb{R}^{2}$ . Let$L$ be any line in$\mathbb{R}^{2}$ not through the origin. Let$\mathbf{p},\mathbf{q},\mathbf{r},\mathbf{s}$ be the vector coordinates of the intersections of$P,Q,R,S$ with$L$ . Then$$\set{P,Q;R,S} = \left(\frac{\mathbf{p}-\mathbf{r}}{\mathbf{p}-\mathbf{s}}\right)\left(\frac{\mathbf{q}-\mathbf{s}}{\mathbf{q}-\mathbf{r}}\right),$$ where the ratios of vectors make sense because they all in the direction of$L$ , and we define$\frac{\lambda\mathbf{v}}{\mathbf{v}} = \lambda.$
- A triangle
$\Delta PQR$ in$\mathbb{P}^{n}$ is a set of three distinct points,$P,Q,R\in\mathbb{P}^{n}$ and the "sides" of the triangle, which are three lines spanned by the three paris of points. - Two triangles
$\Delta PQR$ and$\Delta P'Q'R'$ in$\mathbb{P}^{n}$ are said to be in persepctive from a point$\mathcal{O}$ if the lines$\langle P,P'\rangle, \langle Q,Q'\rangle, \langle R,R'\rangle$ all intersect in a common point$\mathcal{O}$ . - Two triangles
$\Delta PQR$ and$\Delta P'Q'R'$ with sides$p,q,r$ and$p',q',r'$ in$\mathbb{P}^{n}$ are said to be in perspective from a line$L$ if the points$p\cap p',q\cap q',r\cap r'$ all lie on a common line$L$ .
- A projective line in
$\mathbb{P}^{2}$ is the image of a plane through 0 in$\mathbb{R}^{3}$ under the canonical projection$p:\mathbb{R}^{3}\setminus\set{0}\to\mathbb{P}^{2}$ that sends$(x_{0},x_{1},x_{2})$ to the line it spans.
- Any two distinct lines in
$\mathbb{P}^{2}$ intersect at a point. (This is because let$L = \rho(\Pi)$ and$L' = \rho(\Pi')$ be two distinct projective lines. Then$\Pi = \Pi'$ and therefore$\Pi\cap\Pi'$ is a line$l$ in$\mathbb{R}^{3}$ through 0 and$\rho(\Pi)\cap\rho(\Pi') = \rho(\Pi\cap\Pi') = \rho(l)$ , which is by definition a point in$\mathbb{P}^{2}$ .) - Whenever
$\mathbb{F}_ {q}$ is a field, (i.e.$q$ is a prime power), the projective plane$\mathbb{P}^{n}(\mathbb{F}_ {q})$ has$q^{n}+q^{n-1}+...+1$ points and$q^{n}+q^{n-1}+...+1$ lines. Each line contains$q+1$ points and each point is contained in$q+1$ lines. - 3 distinct lines in
$\mathbb{P}^{n}$ that intersect in pairs are either concurrent (i.e. they have a common point) or coplanar.
- If
$\Delta PQR$ and$\Delta P'Q'R'$ are two distinct triangles in$\mathbb{P}^{n}$ which are in perspective from a point, then they are also in perspective from a line.
- Let
$L,L'\subset\mathbb{P}^{2}$ be distinct projective lines. Let$P,Q,R\in L\setminus L', P', Q', R'\in L'\setminus L$ be distinct points. Then the intersection points$$A = \langle P,Q'\rangle\cap\langle P',Q\rangle,\quad B = \langle P,R'\rangle\cap\langle P',R\rangle,\quad C = \langle Q,R'\rangle\cap\langle Q', R\rangle$$ are colinear.