I've found a better way that allows you to get the distance between the point and the line. First, check if the segment is longer than distances between each end and the point (i.e. the projection of the point on the segment line is in the segment) Then, we need to compute the area of the triangle between the ends of the segment and the point. It can be easily done using Heron formula (check wikipedia for details) Then you just have to double the area and divide it with the segment length and you have the distance between the point and the line. You can now check if the point is near enough in your context to be considered on the line.
For instance if you wanted to minimize cos(x) = c1 where c1 is a constant, using gradient descent one way or another yields you that c1 = 0, but the constant term in the taylor expansion of cos(x) is 1 since cos(x) = 1 - x^2/2 + ... This means that you have to include at least the 2nd term for this to work, or even a higher degree depending on the function other than cos(x) in the example.
when the ray casted from the point crosses a vertex, the one intersection is counted twice (because 2 edges are defined to have that point), which will give wrong answers
Thanks so much for this! I needed to find centroids of irregular polygons for a Matter.js project and your explanation and code examples got me up and running quickly.
Out of curiosity, is there a known or best-guess optimal or near-optimal value for the padding in the algorithm? Perhaps related to the mean distance between the sites?
No, I couldn't find an easy, step-by-step algorithm for building Voronoi diagrams (unlike Delaunay triangulation algorithms, which are easy to find). That's why I created this video.
Beautiful and very well made video, I personally loved the old tv vibe to this, not to disregard the instructive yet nicely explained method of gradient descent. Subscribed
Gradient descent is finding optimal minimum point of the function f(x), not finding solution of f(x)=0. However, optimal point of any f(x) is exactly the solution of f'(x) (derivative function of f(x)). So, in case your function has only one variable, to find the solution of f(x)=0, you can replace the derivative term with f(x) and so on. If your function has more than one variable, you can't replace, cause there's only one function has been given, so you do not know that function is depends on which variable (as mentioned above, if you have one variable, f(x) is derivative function depends on x when you use Gradient Descent to find solution). So, the solution is using Least Square Approximation method as the video has shown. Function f^2(variable) always has optimal minimum point. If minimum point's value is 0, it is the solution. If not, GD still finds optimal minimum point, but it is not the solution.
Yes, solving the equation x^5 + x = y for x in terms of y is much more complex than solving quadratic equations because there is no general formula for polynomials of degree five or higher, due to the Abel-Ruffini theorem. This means that, in general, we can't express the solutions in terms of radicals as we can for quadratics, cubics, and quartics. However, we can still find solutions numerically or graphically. Numerical methods such as Newton's method can be used to approximate the roots of this equation for specific values of y. If we're interested in a symbolic approach, we would typically use a computer algebra system (CAS) to manipulate the equation and find solutions.
AWESOME Video! Thanks! Trying to put some basic understanding on this: "We seek a cubic polynomial approximation (ax^3 + bx^2 + cx + d) to cosine on the interval [0, π]." Let's say you want to represent the cosine function, which is a bit wavy and complex, with a much simpler formula-a cubic polynomial. This polynomial is a smooth curve described by the equation where a, b, c, and d are specific numbers (coefficients) that determine the shape of the curve. Now, why would we want to do this? Cosine is a trigonometric function that's fundamental in fields like physics and engineering, but it can be computationally intensive to calculate its values repeatedly. A cubic polynomial, on the other hand, is much simpler to work with and can be computed very quickly. So, we're on a mission to find the best possible cubic polynomial that behaves as much like the cosine function as possible on the interval from 0 to π (from the beginning to the peak of the cosine wave). To find the perfect a, b, c, and d that make our cubic polynomial a doppelgänger for cosine, we use a method that involves a bit of mathematical magic called "least squares approximation". This method finds the best fit by ensuring that, on average, the vertical distance between the cosine curve and our cubic polynomial is as small as possible. Imagine you could stretch out a bunch of tiny springs from the polynomial to the cosine curve-least squares find the polynomial that would stretch those springs the least. Once we have our cleverly crafted polynomial, we can use it to estimate cosine values quickly and efficiently. The beauty of this approach is that our approximation will be incredibly close to the real deal, making it a nifty shortcut for complex calculations.
Thank you for the video!! Took some time to grasp the second example. No surprise. This gradient descent optimization is at the heart of machine learning.
Bro this video is so fire. I get so annoyed by the voices in my actual school videos that they make you watch and this is a huge step up from that it actually makes this seem like it’s a top secret information like you’re debriefing the first nuclear tests or something
