Gauss's magic shoelace area formula and its calculus companion

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Recommends VPfB · 2017-07 · Stack Overflow · on “Calculate the area of intersection of two rotated rectangles in python”

There is a nice expalanation of a polygon area computation here: [this video] Together with links and hints from the linked duplicate question (areas: intersection = polygon1 + polygon2 - aggregated polygon) the missing part is to convert your data to (x,y) coordinates of vertices. i.e. some easy sine and cosine calculations.

Recommends user848964 · 2021-03 · Mathematics Stack Exchange · on “Finding the area of my lake”

Something that might interest you is the shoelace formula [LINK] for the area of a polygon given the cartesian coordinates of the vertices. According to the wikipedia page: [The shoelace formula] can be verified by dividing the polygon into triangles, and can be considered to be a special case of Green's theorem. Mathologer has a [this video] on this topic. His videos are a delight to watch and I always learn something new when watching them, so I recommend checking that out. As for applying the formula to your situation, we would like to choose an origin and set up a cartesian coordinate system. One way to do this is using a program like photoshop which displays the cartesian coordinates of the

Cites PrincessEev · 2018-12 · Mathematics Stack Exchange · on “Derivation of Area Formula in Coordinate Geometry”

This is actually a specific case of something known as Gauss' shoelace formula or Gauss' area formula, which can be generalized beyond a triangle to any number of vertices (under the right circumstances). This is typically presented in a different way than your formula, but they're essentially the same. Let me show how they're the same first: let us have the points $(x_1,y_1), (x_2, y_2), (x_3, y_3)$ defining a triangle. For Gauss' formula, we line these up in a table (see below), with the first point at the bottom and the top, with the points listed going counterclockwise from the first point. From the top-left entry, we connect to the entry one down and one to the right with a blue line, u

Cites Clemens Bartholdy · 2021-08 · Mathematics Stack Exchange · on “Which smooth closed curve has the highest area to perimeter ratio?”

I was thinking of what way to loop a string of a given length such that area inside is maximum. The reason I mentioned smooth in the question is because if we had a mathematical string, then we could deform tiny segments making bumps on the perimeter and reach an infinite perimeter. My attempt: Let's assume the curve to be $\gamma(t)$ and let the parameter domain be $\left[ 0, t_{max} \right]$ then the arclength is given as: $$ L = \int_0^{t_{max} } \sqrt{ \dot{\gamma} \cdot \dot{\gamma} } dt $$ Note: The above arclength is fixed. The area of the parametric curve maybe given by gauss's shoe lace formula [this video] in vector form: $$ A= \int_0^{t_{max} } \gamma \times \dot{\gamma} dt$$ And, now,

Cites Herbert Schreib · 2021-12 · Stack Overflow · on “How can I calculate the area of a bezier curve?”

Couldn't you use an application of Gauss's magic shoelace theorem by getting a set of data points by changing T, then simply inputting that into the equation? Here's a simple video demo [this video] And then here's the wiki [LINK]

Cites BBSysDyn · 2023-07 · Stack Overflow · on “Calculate area of polygon given (x,y) coordinates”

Shoelace formula comes from computing internal triangles based on consecutive points around the polygon. It could be more informative to base the code on this explanation. The area of polygon is simply length of cross product (determinant) of its two neighboring vectors divided by two, all added up, [this video] import numpy as np, numpy.linalg as lin def area(pts): ps = np.array([0.5 * lin.det(np.vstack((pts[i], pts[i+1]))) for i in range(len(pts)-1)]) s = np.sum(ps) p1,p2 = pts[-1],pts[0] # cycle back, last pt with the first s += 0.5 * lin.det(np.vstack((p1,p2))) return np.abs(s) points = np.array([[0,0],[10,0],[10,10],[0,10]]) area(points) # 100