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Hi Colin thank you for making this place so interesting. I struggle to teach myself maths but greatly enjoy it. As I see it the most applicable branch of maths

by Dn_Ab 15y ago

Hi Colin thank you for making this place so interesting. I struggle to teach myself maths but greatly enjoy it. As I see it the most applicable branch of maths for games falls under the umbrella of Clifford/Geometric Algebra - stuff like quaternions, cross products and bivectors. Indeed graphics programmers would gain a lot of power and elegance from the unified view provided by geometric algebra - rather than the hodgepudge of seemingly unrelated concepts as it is currently presented. Topics covered in a linear algebra class that are typically used in games are basic - vector arithmetic, dot products, linear and affine transformations and projection. Below you note that a vector and a matrix are different things. But I think the word is polymorphic. Sometimes, the notation aspect - table based syntax for arranging numbers is the focus. So someone will say that matrices are made up of n column vectors. Other times the focus will be on the linear transformation aspect. Where a matrix is more like a higher order function and is evidently different from the entities it is transforming. It simply requires a change of frame.

ColinWright 15y ago
The ability to move freely between different conceptualizations is both important and useful, but if used too early it comes at a cost. Thinking of vectors as elements of the space and matrices as ways to transform the space gives a clear separation of concerns, and can make it easier to grasp what's going on. Later you can discover that by dividing up the matrix the right way you can think of it as 1xN matrices, and then you can associate those with vectors and gain further insight as to what's, but trying to do that early can be very confusing. After all, an int is a kind of number, and a float is a kind of number, and we can move between those, but there are dangers unless you've got the difference clear in your mind. How many people have complained that 5*0.2 isn't 1.0? Knowing what's happening underneath, and then creating the mappings in your understanding seems to be the fastest way to mastery.
- Dn_Ab 15y ago
  Oh right I see. But don't you think introducing matrices as a type of transformation between vector spaces is too much too early? One reason maths is so difficult is that our brain simply has no point of comparison to bootstrap the concepts with. You just get used to it over time with effort. Starting so abstract would leave many people lost at sea. But then again, a lot of why matrix algebra was so confusing at first is it all seemed so arbitrary. Just memorizing a bunch of boring human calculable unmotivated algorithms. I get angry thinking about it. It wasn't till I read about how they came about in 1800s and studied linear algebra did they start to make sense. Maybe maths beyond basic probability,geometry and algebra - which can be readily motivated to everyday life - should not be taught to everyone. Then less people would find it useless and more people would not be so embittered towards the subject.

by Dn_Ab 15y ago

ColinWright 15y ago

The ability to move freely between different conceptualizations is both important and useful, but if used too early it comes at a cost. Thinking of vectors as elements of the space and matrices as ways to transform the space gives a clear separation of concerns, and can make it easier to grasp what's going on. Later you can discover that by dividing up the matrix the right way you can think of it as 1xN matrices, and then you can associate those with vectors and gain further insight as to what's, but trying to do that early can be very confusing. After all, an int is a kind of number, and a float is a kind of number, and we can move between those, but there are dangers unless you've got the difference clear in your mind. How many people have complained that 5*0.2 isn't 1.0? Knowing what's happening underneath, and then creating the mappings in your understanding seems to be the fastest way to mastery.

Dn_Ab 15y ago
Oh right I see. But don't you think introducing matrices as a type of transformation between vector spaces is too much too early? One reason maths is so difficult is that our brain simply has no point of comparison to bootstrap the concepts with. You just get used to it over time with effort. Starting so abstract would leave many people lost at sea. But then again, a lot of why matrix algebra was so confusing at first is it all seemed so arbitrary. Just memorizing a bunch of boring human calculable unmotivated algorithms. I get angry thinking about it. It wasn't till I read about how they came about in 1800s and studied linear algebra did they start to make sense. Maybe maths beyond basic probability,geometry and algebra - which can be readily motivated to everyday life - should not be taught to everyone. Then less people would find it useless and more people would not be so embittered towards the subject.