Analytic Geometry
Table of Contents
1. Cartesian Coordinate System
- Cartesian Space: Space with coordinates attached.
- Cartesian coordinates specify the point in an \(n\)-dimensional Euclidean space with \(n\)-tuple of numbers which is uniquely determined by the affine points of the orthogonal basis.
1.1. Distance
- The Euclidean distance in a Cartesian coordinates is given by the Pythagoras's theorem.
2. Barycentric Coordinate System
2.1. Definition
- Given \( n+1 \) points \( \{A_i\}_{i=0}^n \) in a \(n\)-dimensional affine space that are affinely independent, the barycentric coordinate \( (a_0:\mathord{\dots}: a_n) \) can be constructed, such that: \[ (a_0+\cdots + a_n)\overrightarrow{OP} = a_0\overrightarrow{OA_0} + \cdots +a_n\overrightarrow{OA_n}. \]
2.2. Normalized Barycentric Coordinates
- Absolute Barycentric Coordinates
- With the additional condition: \[ \sum a_i = 1. \]
3. Pappus's Centroid Theorem
3.1. First Theorem
- An area of the surface generated by rotating a plane curve \( C \) around an external axis in the same plane is \[ A = sd \] where \( s \) is the arc length and \( d \) is the distance the centroid of \( C \) travels.
3.2. Second Theorem
- An volume of the solid of revolution generated by rotating a plane figure \(F\) around an external axis in the same plane is \[ V = Ad \] where \(A\) is the area of \(F\) and \(d\) is the distance the centroid of \(F\) travels.
3.2.1. Examples
- The volume of a torus: \(V_{\text{torus}} = \pi r^2 \cdot 2\pi R = 2\pi^2r^2R\).
4. Envelope
- That weird light at the bottom of a mug — ENVELOPES - YouTube
- Envelope \(E\) of a family of curves \(F(x,y,t)=0\) is tangent to the curves everywhere.
- A family of curves does not necessarily has an envelope, and may have multiple envelopes.
4.1. Envelope Conditions
- \[ F(x,y,t)=0 \]
- \[
\frac{\partial F(x,y,t)}{\partial t}=0
\]
- One can also think of it as the limits of intersections.
- \[ \lim_{u\to t}\frac{F(x,y,t)-F(x,y,u)}{t-u}=\lim_{u\to t}\frac{0}{t-u} \]
- One can also think of it as the limits of intersections.
- \(E(x,y)=F(x,y,t(x,y))\) that satisfies the envelope conditions.
5. Stereographic Projection
Bijective map between \( \mathbb{R}^n \) and \( S^n \backslash \{ \text{north pole} \}\). It is often given by: \[ (X_1, X_2, \dots) \in \mathbb{R}^n\mapsto \left( \frac{2X_1}{1+X_1^2 + X_2^2 + \cdots} , \frac{2X_2}{1+X_1^2 + X_2^2 + \cdots}, \dots, -\frac{1- X_1^2 - X_2^2 - \cdots}{1+X_1^2+X_2^2 + \cdots}\right) \in S^n \subset \mathbb{R}^{n+1}. \] Graphically, this maps a point \( P \) on a plane to the point \( Q \) on the sphere, which is centered at the origin with radius 1, by finding the intersection point \( Q \) between the line connecting \( P \) and the north pole \( (0,0,\dots, 1) \), and the sphere.
The position of the sphere may differ by author.
6. Lemniscate
6.1. Lemniscate of Bernoulli
6.1.1. Equations
Cartesian \[ (x^2+y^2)^2 = a^2(x^2-y^2) \]