Math, Linear Algebra

  • Essence of Linear Algebra .

    • Very good.

    • Watched everything, but skipped the eigenvector and eigenvalues part, as it wasn't relevant for what I'm doing.

Matrix

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    • Maps 2 dimensions to 3 dimensions.

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    • Maps 3 dimensions to 2 dimensions.

  • Matrix multiplication :

    • Matrices give us a language to describe transformations, where the columns represent those coordinates.

    • Matrix multiplication is just a way to compute what this transformation does to a given vector.

    • Any matrix can be interpreted as a transformation of space.

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  • Matrix composition :

    • .

      • i (green)  will point to 1, 1 .

      • j (red)  will point to -1, 0 .

      • This constitutes a Rotation + Shear.

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      • First apply the rotation, and then apply the shear.

      • This is the same as the composition 'rotation + shear'.

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      • A composition has the same meaning as applying one transformation, then another.

      • This is read from right to left: the first transformation is Rotation, then Shear.

      • This comes from function notation f(g(x)) .

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      • This is true, as you are just applying C, then B, then A, either way.

  • Determinant :

    • How much the AREA has scaled.

      • Or VOLUME if in 3D; imagine a 1x1x1 cube.

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      • If the determinant is zero, it means that it squashes everything into a lower dimension.

    • If the determinant is negative, it means that the orientation is flipped (the space turns on itself) (a sheet of paper flips).

    • Calculation :

      • 2D:

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        • Proof: .

      • 3D:

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        • Proof: 'It's complicated and doesn't really make sense as the "essence of linear algebra"'.

  • System of equations :

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    • "Rank": number of dimensions in the output of a transformation.

    • "Column Space": Set of all possible outputs Av

  • Inverse Matrix :

    • Does the inverse of a transformation.

      • If a matrix rotates 90º clockwise, the inverse of this would be rotating 90º counter-clockwise.

    • .

      • Doing the transformation A, and then doing the inverse of A, it's the same as not doing anything; hence the Identity Matrix.

    • OpenGL Normal Mapping Quote:

      • Note that we use the transpose function instead of the inverse function here. A great property of orthogonal matrices (each axis is a perpendicular unit vector) is that the transpose of an orthogonal matrix equals its inverse. This is a great property as inverse is expensive and a transpose isn't.

  • Dot Product :

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    • Usually used to analyze the angle between two vectors:

      • a.b > 0 : angle less than 90º.

      • a.b = 0 : angle equal to 90º.

      • a.b < 0 : angle greater than 90º.

      • If the value is positive, it means the vectors point "in the same direction".

  • Cross Product :

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    • Calculating:

      • The Cross Product is just the determinant of the matrix, as the determinant computes the AREA of the matrix.

        • Also, consider the orientation.

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        • This is a mathematical trick to "help" calculate the determinant above.

    • Orientation:

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  • Eigenvector and Eigenvalues :

    • Eigenvector: A vector that remains in its own span.

    • .

Quaternions

  • Zero rotation :

    • (0, 0, 0, 1) .

  • Quaternion multiplication :

    • Is rotation composition.

    • In the context of 3D rotations, multiplication is the only operation that directly corresponds to combining or applying rotations.

    • Non-commutativity matters: q2q1≠q1q2 . This matches the non-commutativity of rotation matrices.

  • Quaternion conjugate / inverse :

    • The conjugate of a unit quaternion is its inverse.

    • Needed when rotating vectors: $qvq^{−1}$.

    • Useful for undoing a rotation.

  • Quaternion scalar multiplication :

    • Not used for representing rotations, since only unit quaternions encode valid rotations.

  • Quaternion addition :

    • Is averaging rotation.

    • ChatGPT:

      • Addition of quaternions has no direct geometric meaning for rotations.

      • It can be used in interpolation schemes (e.g., normalized linear interpolation), but addition itself is not a “rotation operation”.

  • Why Use Quaternions in Graphics :

    • Numerical stability : avoids gimbal lock (unlike Euler angles).

    • Compactness : 4 numbers vs 9 in a matrix.

    • Efficient interpolation : spherical linear interpolation (slerp) is defined directly on quaternions, which is critical for animation blending.

    • Fast composition : quaternion multiplication is cheaper than matrix multiplication.

  • Euler to Quaternion :

    • quaternion_from_euler_angles() .

    • Let:

      • roll = α (rotation around X)

      • pitch = β (rotation around Y)

      • yaw = γ (rotation around Z)

    • Assume intrinsic rotation order Z → Y → X  (common convention in graphics).

    • Quaternion (w, x, y, z): 

      w = cos(α/2) * cos(β/2) * cos(γ/2) + sin(α/2) * sin(β/2) * sin(γ/2)
x = sin(α/2) * cos(β/2) * cos(γ/2) - cos(α/2) * sin(β/2) * sin(γ/2)
y = cos(α/2) * sin(β/2) * cos(γ/2) + sin(α/2) * cos(β/2) * sin(γ/2)
z = cos(α/2) * cos(β/2) * sin(γ/2) - sin(α/2) * sin(β/2) * cos(γ/2)

    • The above assumes Z-Y-X order  (yaw → pitch → roll). If your engine uses a different Euler order, the formulas change.

    • Always normalize the quaternion after conversion from Euler to keep it valid.

  • Quaternion to Euler : 

    pitch (β) = asin( 2 * (w*y - z*x) )
yaw   (γ) = atan2( 2 * (w*x + y*z), 1 - 2*(x*x + y*y) )
roll  (α) = atan2( 2 * (w*z + x*y), 1 - 2*(y*y + z*z) )

    • The above assumes Z-Y-X order  (yaw → pitch → roll). If your engine uses a different Euler order, the formulas change.

    • atan2  is the 2-argument arctangent, which handles quadrant correctly.

    • Gimbal lock happens if pitch = ±90°. In that case, roll and yaw are not uniquely defined.

  • ~ Understanding Quaternions for Game Dev .

    • Critiques the explanations by 3blue1brown and numberfy for complicating the concept by introducing 4D spheres.

    • His explanation is based on defining a rotation axis, etc.

    • His intent to simplify the process is fine, but there is no foundational reasoning; it feels like rote memorization.

    • For graphics programming, this is probably the only explanation that makes sense to look at; 3blue1brown's is somewhat irrelevant, curiously.

  • ~ Quaternion and 3D Rotation .

    • Yeah, nothing useful for computer graphics could be understood.

  • ~ Visualization of Quaternions as a 4D sphere .

    • Only provides several images to help visualize the 4D sphere, nothing more.

    • Yeah, nothing useful for computer graphics could be understood.

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Etc

Spherical Coordinates

Polar Coordinates

Wave Function Collapse (WFC)