# Geometric Drawings

Sketching with Math and Quasi Physics

## Lines

Two lines, $$a_1x+by_1+\;c_1=0$$ and $$a_2x+by_2+\;c_2=0$$ intersect if $$a_1b_2\neq a_2b_1$$

The point of intersection is

$$x=(b_1c_2-b_2c_1)/(a_1b_2-a_2b_1)$$

$$y=(a_1c_2-a_2c_1)/(a_1b_2-a_2b_1)$$

## Bézier Curve

### Linear Bézier curves

A linear Bézier curve is simply a straight line between two points, which can be defined as linear interpolation between two points

$$\mathbf {B}_{linear\mathbf {P} _{0}, \mathbf {P} _{1}} (t)=(1-t)\mathbf {P} _{0}+t\mathbf {P} _{1}{\mbox{ , }}0\leq t\leq 1$$

A quadratic Bézier curves can be defined as linear interpolation between corresponding points on two linear Bézier curves.

$${\displaystyle \mathbf {B}_{quadratic \mathbf {P} _{0}, \mathbf {P} _{1}, \mathbf {P} _{2}} (t)=(1-t)\mathbf {B}_{linear \mathbf {P} _{0}, \mathbf {P} _{1}}(t)+t\mathbf {B}_{linear \mathbf {P} _{0}, \mathbf {P} _{2}}(t){\mbox{ , }}0\leq t\leq 1}$$

### Qubic Bézier curves

And a cubic Bézier curves is linear interpolation between corresponding points on two quadratic Bézier curves.

$$\mathbf {B}_{cubic{\mathbf {P} _{0}, \mathbf {P} _{1}, \mathbf {P} _{2},\mathbf {P} _{3}}} (t)=(1-t)\mathbf {B} _{quadratic \mathbf {P} _{0},\mathbf {P} _{1},\mathbf {P} _{2}}(t)+t\mathbf {B} _{quadratic \mathbf {P} _{1},\mathbf {P} _{2},\mathbf {P} _{3}}(t)$$

$$=(1-t)^{3}\mathbf {P} _{0}+3(1-t)^{2}t\mathbf {P} _{1}+3(1-t)t^{2}\mathbf {P} _{2}+t^{3}\mathbf {P} _{3}{\mbox{ , }}0\leq t\leq 1$$

## B-Spline Curve

$$P$$ is a set of control points and $$t$$ is a vector of non-decreasing numbers called "knot vector" which has $$number\;of\;control\;points\;+\;order\;of\;the\;curve(n)\;+\;1$$ elements, e.g. $$(0, 0, 0, 1, 2, 3, 3, 3)$$.

B-spline basis function, $$B$$ is defined on the knot vector and used to weight the control points.

$${\displaystyle Spline_{n,t}(x)=\sum _{i}P _{i}B_{i,n}(x)}$$

$$B_{i,0}(x):=$$

$$\begin{array}{lc}1&\mathrm{if}\; {t}_{i}\leq{x}<{t}_{i + 1}\end{array},$$

$$\begin{array}{lc}0&otherwise\end{array}$$

$${\displaystyle B_{i,k}(x):={\frac {x-t_{i}}{t_{i+k}-t_{i}}}B_{i,k-1}(x)+{\frac {t_{i+k+1}-x}{t_{i+k+1}-t_{i+1}}}B_{i+1,k-1}(x)}$$

B-spline Basis Functions: Definition

B-spline curve

## Circumcircle and Circumcenter

Circumcircle
The circle that passes through three vertices of a triangle.
Circumcenter
The center of the circumcircle. The intersection of the perpendicular bisectors of a triangle.

## Incircle and Incenter

Incircle
The circle tangent to the three sides of a triangle.
Incenter
The center of the incircle. The intersection of angle bisectors of a triangle.

## Excircles and Excenters

Excircles
The circles tangent to one of a triangle's sides and to the extensions of the other two.
Excenters
The centers of an excircles. Points where the external angle bisectors of a triangle intersect.

## Orthocenter

Orthocenter
The intersection of the three altitudes of a triangle.

## Centroid

Centroid
The arithmetic mean position of all the points in a shape. The centroid of a triangle is the intersection of its three medians.

## Reuleaux Triangle

A reuleaux triangle can rotate within a square while touching all four sides of a square.

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