Handy Math
Sketching with Math and Quasi Physics
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Sine, Cosine and Tangent
$$\mathrm{sine}=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$$
$$\mathrm{cosine}=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$$
$$\mathrm{tangent}=\frac{\mathrm{sine}}{\mathrm{cosine}}$$
* Note that p5.js is based on the screen coordinate system in which y increase from top to bottom. The opposite is flipped by multiplying -1 in the rendering below.
Rotating 2D points
$$\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}$$
Learn more
A Gentle Primer on 2D Rotations
Euclidean distance between two points
Euclidean distance is the most intuitive, "ordinary" straight-line distance between two points in Euclidean space.
Two dimensions
$$p=(p_x,p_y),\;q=(q_x,q_y)$$
$$\mathrm{distance}(p,q)=\sqrt{(p_x-q_x)^2+(p_y-q_y)^2}$$
n dimensions
$$p=(p_1,p_2\cdots,p_n),\;q=(q_1,q_2\cdots,q_n)$$
$$\mathrm{distance}(p,q)=\sqrt{(p_1-q_1)^2+(p_2-q_2)^2+\cdots+(p_n-q_n)^2}=\sqrt{\sum_{i=1}^n(p_i+q_i)^2}$$
Manhattan distance between two points
Euclidean distance is not the only way to measure distance between two points. In mathmatics, distance is defined by a function called distance function or metric, which can be any function that satisfy a set of conditions.
Manhattan distance is an example of a non-Euclidean distance, which is the sum of the absolute differences of their Cartesian coordinates, or the distance a car would drive in a city laid out in square blocks.
Two dimensions
$$p=(p_x,p_y),\;q=(q_x,q_y)$$
$$distance\left(p,\;q\right)\;=\;\left|p_x\;-\;q_x\right|\;-\;\left|p_y\;-\;qy_{}\right|\;$$
Perspective Projection (Fixed Camera)
\(\lbrack x,y,z\rbrack\) projects to \(\left[x\frac{\displaystyle d_1}{\displaystyle d_2},y\frac{\displaystyle d_1}{\displaystyle d_2}\right]\)
- \(d_1=\) Focal length
- the axial distance from the camera center to the image plane
- \(d_2=\) Subject distance
- the axial distance from the camera center to the subject
Dot Product
Two dimensions
$${\boldsymbol v}_\mathbf1=\lbrack x_1,y_1\rbrack,\hspace{8px}{\boldsymbol v}_\mathbf2=\lbrack x_2,y_2\rbrack$$
$${\boldsymbol v}_\mathbf1\cdot{\boldsymbol v}_\mathbf2=x_1x_2+y_1y_2$$
n dimensions
$$\boldsymbol a=\lbrack a_1,a_2,a_3,\dots,a_n\rbrack,\hspace{8px}\boldsymbol b=\lbrack b_1,b_2,b_3,\dots,b_n\rbrack$$
$$\boldsymbol a\cdot\boldsymbol b=a_1b_1+a_2b_2+a_3b_3\dots a_nb_n$$
- Two non-zero vectors a and b are orthogonal if and only if a ⋅ b = 0.
- If the dot product is positive, the angle between the vectors is less than 90°.
- If the dot product is negative, the angle between the vectors is more than 90°.
Cross Product
3 dimensions
$${\boldsymbol v}_\mathbf1=\lbrack x_1,y_1,z_1\rbrack,\;{\boldsymbol v}_\mathbf2=\lbrack x_2,y_2,z_2\rbrack$$
$${\boldsymbol v}_\mathbf1\times{\boldsymbol v}_\mathbf2=\lbrack y_1z_2-z_1y_2,z_1x_2-x_1z_2,x_1y_2-y_1x_2\rbrack$$
- The cross product is orthogonal to both of the original vectors(v1 and v2), which implies the cross product is perpendicular to the plane defined by the original vectors.
- If the cross product of two vectors is the zero vector, either one or both of the inputs is the zero vector, or they are parallel or antiparallel.