parametric cubic curve
boundary representation
blending function
spline
b-spline
cubic bezier curve
de casteljau algorithm
I have been diving into these topics since few days and I have not understood anything. I tried peter shirley's textbook on Computer Graphics. Foley et al, Hearn baker etc...
It is clear to me that I lack the mathematics preriquisites for understanding this. Thus, I am deeply asking for resources that I can do to brush up. Me being a computer engineering graduate I have done math classes, it is just that I became like a calculator instead of understanding the logic behind doing the formulas and steps so need a short revision.
I am not very knownledgeable on Computer Graphics Maths, but i have an interest in Texture Synthesis so i might be able to help as i was not mathematically literate when i started. Here's my strategy :
1 - Find a recent research paper of low-medium complexity 2 - Read it in whole 3 - take note of any term i don't understand 4 - Read the research paper where this idea was introduced 5 - Repeat steps 3 - 5
I am currently reading a fascinating 1973 paper about how different statistics affect our ability to differenciate between different textures. I am not a pro of Texture Synthesis maths but i have a good idea of where the field started, where it is going and understand the decisions of modern papers much better.
Here is one excellent resource for all of the things you mentioned: Introduction to Computing with Geometry by Dr. C.-K. Shene https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/
When I was implementing these, the key was to visualise everything. It was much easier to see if things are correct by stepping through the algorithms graphically than just by looking at the math.
Sounds like you’re interested in curve fitting. I would recommend intro to statistical learning for a good intro to related topics.
Free download
https://www.statlearning.com/
Oh wow! I had no idea they came out with a Python version!
Everything is basically calculus and linear algebra under the hood (until it's not, at which point you'll know)
Take some upper level physics courses. It's all about deriving equations and models rather than just using formulas.
Math is a language with many dialects, so you should treat it like learning a language in that you both have to start simple and build up, and use it. Take the time to learn what the symbols are saying for whatever dialect you are using.
It sounds like you're getting into geometric kernel programming for something like CAD. Go brush up on parametric equations, implicit vs explicit representations, and play around with simple spline examples. 2D is enough to get the idea. You'll need to know some basic calculus for this, as splines are usually constrained by the derivative at the node. You can get quite far with just Wikipedia, YouTube videos, some thinking, and some exploratory programming. Hell, throw AI in there as well.
A big part of why I'm saying this is because at their core, these are not difficult equations from a computational perspective. You really just need to understand addition, subtraction, multiplication, and division. My point is that the hard part about math often isn't the calculation part.
Let's take B-Splines for an example and look at what Wiki says
Either this is simple or incomprehensible, so let's break it down.What does piecewise mean? Here's an example piecewise function.
Given you're on HN and looking at computer graphics, I take it you're familiar with if statements? That's the same thing here. Let's translate. Looking at things from a different perspective can really help. We should think of x as a location and we're just describing something about that location. Here's another piecewise function, but now in everyday English: "Across the street all the houses end in even numbers, but on my side of the street all the houses are odd." There's always more ways to look at something. If you can't see what something means from this angle, try another. Be that another book teaching the same thing or another framework to describe the same thing. Get multiple perspectives. One will make more sense than another but which one that is tends to be personal."polynomial of order n"
A polynomial is an equation like a_0 + a_1 x + a_2 x^2 + ... + a_n x^{n}. Sorry, let me rewrite that: a_0 x^0 + a_1 x^1 + a_2 x^2 + ... + a_n x^{n}
Now in code
But we should break this down more, just like before. What is a_0? It is just a number. What does this represent? A point. What is a_1*x? That's a line. x is an arbitrary location. The a_1 is a scaling factor. So if a_1 is the number 2, then what our x is representing is all numbers (..., -4, -2, 0, 2, 4, ...} (is that all my neighbors across the street?) The a_2 x^2 is a curve, specifically a parabola. We can do the same thing. But also remember we're summing them together. So you have a point, a line, and a parabola. When you add them together you get something that is none of those things.What is really important here is that these functions you are interested in have a really critical property. They can be used to approximate almost anything. That almost part is really important, but you're going to have to dig deeper.
My point here is that take time to slow down. Don't rush this stuff. I promise you that if you take time to slow down then the speed will come. But if you try to go too fast you'll just end up getting stuck. This is tricky stuff. When learning it doesn't always make a ton of sense and unfortunately(?) when you do understand it it is almost obvious. Think of the slowing down part like making a plan. In a short race you should just take off running without thinking. But if it is longer then you will go faster if you first plan and strategize. It is the same thing here and I promise you this isn't a short race. You don't win a marathon by winning a bunch of consecutive sprints. The only thing you win by trying that is a trip to the hospital.
So slow down. Break problems apart. Find one part you think is confusing and focus on that. If it all seems confusing then try to walk through and force yourself to say why it is confusing. Keep doing this until you have something you understand. It is an iterative process. It'll feel slow, but I promise it pays dividends. Any complex problem can be broken down into a bunch of small problems[0]
You got this!
[0] This sentence doesn't just apply to how to figure things out, it applies directly to what you're trying to do and why you want these functions. If it doesn't make sense now, revisit, it will later.
Very nice explanation!
In particular; the idea of breaking down a polynomial as a sum of terms, mapping each term to a graphical view (i.e. analytic to coordinate geometry) and then realizing such a sum of terms can be a complex curve (i.e. a complex graphical view) which can be an approximation of almost any function.
The teaching of mathematics has become so abstract that students are not taught how to map it to geometry which is THE way to build intuition. You can understand a lot by just imagining 2D/3D mappings before generalizing to n-dimensional vector spaces. Every student/beginner should study I.M.Gelfand's Functions and Graphs so that they can train themselves to imagine the graphs corresponding to a string of symbols.
For Splines I found Christian Reinsch's classic paper to be a really good resource. You can find it various places on the web i.e here: https://tlakoba.w3.uvm.edu/AppliedUGMath/auxpaper_Reinsch_19...
The algorithm at the end is written in Algol but I found it pretty easy to understand.
It's a bit difficult to say, since some of those textbooks (e.g. Shirley) cover preliminary math topics as well. It sounds like you need to get a stronger understanding of calculus and linear algebra, and approach maths from a healthier viewpoint. Something like Axler's Linear Algebra Done Right often forces people to treat linear algebra with the slow care it requires, but this may be a bit too extreme.
Now that you have some applications in mind, write down all of the terms and concepts that are involved that you don't understand. Go back to a first-year textbook, something like Stewart's Calculus is probably ideal since it is designed to work up to e.g. Bezier curves. However, this time, I would not suggest relying on the exercises for understanding. Instead, place strong emphasis on the definitions and theorems, drawing them together into a diagram with the terms from earlier to really understand why each concept is required for another concept. Your objective should be to see how these ideas lead to the definition of those curves, and why it has to be that way. Be skeptical and try thinking of alternatives; you might find them to have issues, or coincide with what has already been found! Make sure you understand each concept first before you apply it; this will not be a fast process, and if you need to check something, be sure to do it. But thinking in the context of what you want to learn later is often helpful, I find.
There are other, more detailed guides out there, mostly geared toward higher level math, e.g. https://www.susanrigetti.com/math and you might find the early parts of these useful too. Also, I have found chatbots to be reasonable at giving tailored explanations for a given topic, so there is rarely harm in using them to supplement your learning. Just don't use them to solve problems for you.
I love Axler, but it is absolutely the wrong book for someone only interested in the applications of linear algebra. (Strang is usually recommended in this case but I haven’t read it.)
Have you checked the curriculum of university program? Quick research shows MIT has stuff available. It might seem like a long shot but if you feel like you lack foundation knowledge, it might be worth looking over some of the material.
It sounds like you've got something specific in mind when you say, "modeling". The term modeling is used in a lot of different situations to mean different things. For example, it could mean to make a 3d model in Blender, it could mean to pose for someone to paint you or to take a photo, with databases it's used to mean modeling the data, with statistics it's used to mean finding a way to simply represent and reason about the data (create a model of it).
The things you've listed out make me guess you want to write 2d or 3d image rendering software. Is that right?
If that's the case, there's no substitute for trying to recreate certain algorithms or curves using a language or tool that you're comfortable with. It'll help you build an intuition about how the mathematical object behaves and what problems it solves (and doesn't). All of these approaches were created to solve problems, understanding the theory of it doesn't quite get you there. If you don't have a good place to try out functions, I recommend https://thebookofshaders.com/05/ , https://www.desmos.com/calculator , or https://www.geogebra.org/calculator .
A good place to start is linear interpolation (lerp). It seems dead simple, but it's used extensively to blend two things together (say positions or colors) and the other things you listed are mostly fancier things built on top of linear interpolation.
https://en.wikipedia.org/wiki/Linear_interpolation
For bezier curves and surfaces here are some links I've collected over the years: https://ciechanow.ski/curves-and-surfaces/ https://pomax.github.io/bezierinfo/ https://blog.pkh.me/p/33-deconstructing-be%CC%81zier-curves.... http://www.joshbarczak.com/blog/?p=730 https://kynd.github.io/p5sketches/drawings.html https://raphlinus.github.io/graphics/curves/2019/12/23/flatt...
A final note: a lot of graphics math involves algebra. Algebra can be fun, but it also can be frustrating and tedious, particularly when you're working through something large and make a silly mistake and the result doesn't work. I suggest using sympy to rearrange equations or do substitutions and so on. It can seem like overkill but as soon as you save a few hours debugging it's worth it. It also does differentiation and integration for you along with simplifying equations.
https://docs.sympy.org/latest/tutorials/intro-tutorial/intro...
What you are trying to learn/understand falls under the rubric of Scientific Computing/Numerical Methods/Numerical Analysis/Numerical Algorithms. The mathematics underpinning them is quite wide but mostly Linear Algebra and Calculus. You might find the following useful for your study;
1) Scientific Computing by Michael Heath - Classic text covering a broad swath of domains and tries to build motivation/intuition before the mathematics (affordable Indian edition available).
2) Mathematical Principles for Scientific Computing and Visualization by Gerald Farin and Dianne Hansford - Nice overview of needed background. The authors also have a book named Practical Linear Algebra: A Geometry Toolbox which you might find a ideal companion.
3) Numerical Methods: Fundamentals and Applications by Rajesh Kumar Gupta - A relatively recent book with a really broad coverage of subjects and detailed mathematical expositions (affordable Indian edition available).
4) Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics by Justin Solomon - Good explanations (affordable Indian edition available). Free ebook available at https://people.csail.mit.edu/jsolomon/
5) Finally, Mathematics for Physicists: Introductory Concepts and Methods by Alexander Altland and Jan Von Delft is an excellent book to have as a reference. It has three sections viz. Linear Algebra, Calculus and Vector Calculus. The presentation is very precise, does not focus on proofs/lemmas but on concepts and covers a wide swath of important mathematics.
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