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Recap and set the stage for the day
Blackboard: Inverting a 2x2 Matrix
Blackboard: Producing the four scalar equations for the inverse of the A matrix
Blackboard: Solve for y by back-substituting
Blackboard: Simplify the term for y
Blackboard: Solve for x, noting how it is a similar term to that for y
Blackboard: Consider taking a shortcut to the solutions for z and w
Blackboard: Solve for w
Blackboard: Solve for z
Blackboard: Call for an algebraic blackboard program and note how the terms have the same divisor, before double-checking the workings
Blackboard: Try multiplying our new matrix by the original one in order to see if we end up with the identity matrix
Blackboard: Double-check the workings for z
Blackboard: Compact our inverse of the A matrix
Blackboard: Mnemonically remembering the determinant of a 2D matrix
Blackboard: Matrix transpose vs matrix inverse
Blackboard: Our actual equation for doing the UV mapping
Run the game and demo the current situation
handmade_render.cpp: Look through how DrawRectangleQuickly() is currently working
Blackboard: Note that we can compute UxVy - UyVx ahead of time, keeping the code otherwise identical
handmade_render.cpp: Make DrawRectangleQuickly() compute the variables according to our equation
Run the game, see that it's not right and investigate why
build.bat: Switch to -Od, break into DrawRectangleQuickly() and inspect the values
handmade_render.cpp: Make DrawRectangleQuickly() correctly compute the nXAxis and nYAxis
Run the game and see that it now works perfectly
A few words on the importance of understanding maths concepts in diagnosing and solving programming problems
Miblo Q: Semi-off-topic: Any chance you could describe a little more what you'd want of an algebraic blackboard program?
Blackboard: Why Casey Doesn't Like Mathematica
thesizik Q: Whoa...
areriff Q: Will there be more things like this that require math like this in further parts of the game? Although it was pretty basic, my math is a bit rusty
bryanwwag Q: Do you consider math to be more of a tool to solve problems or more of a learned art to be mastered?
lkey144 Q: Googling what I believe you wanted turned up: http://reduce-algebra.com/ Automatic and user-controlled simplification of expressions and calculations with symbolic matrices
symbolic_butt Q: I just wanted to note how your intuition of the transform seemed really important at finding the bug, way more important than typing the correct formula. Uh no wait
zilarrezko Q: I actually just learned about determinants in Linear Algebra today. Do you think we will see them later on? Are determinants used often?
gasto5 Q: Do you get annoyed by some symbols meaning different things in different subcategories of math?
cubercaleb Q: What about 3D with an isometric view?
jpmontielr Q: What do you think of Khan Academy as a math learning resource?
cubercaleb Q: Yeah, I mean a 2D game that is drawn in 3D
lkey144 Q: Recognizing "the kinds of things you are working with, and the kind of problem you are working on" is a fundamental skill. I'm not all that sure how to teach that directly, though, beyond doing something like what you are doing here, and / or repeated experience with solving problems
Imagining a series that is more focused on mathematical problem identification and solving