linear

• we will want to represent the geometry of points in space • we will often want to perform (rigid) transformations to these objects to position them – translate – rotate • or move them in an animation – time varying tform • position or move virtual camera • we also may use non-rigid tforms to specify shape – scale an object (uniform scale). – squash a sphere into an ellipsoid (non-unform scale. less often)

SO....

• so we must understand how to manipulate 3d coordinates and transforms • we must pay attention to order of tforms • we must pay attention to the role of the coordinate system w.r.t. which we perform a tform • we will look at linear and affine transformations • at end of the day, our code will have vertices with 3d coords and we will use 4 by 4 matrices to describe properly manipulate them • but to figure out what to code, we need to first do some thinking/paper-pencil work.

Geometric data types

• we describe a point using a coordinate vector

 x   y  z

• specifies position wrt an agreed upon coordinate system – three agreed directions – agreed origin – if we change agreed upon c.s., we must change the coordinate vector • so a point is specified with a coordinate system and a coordinate vector

4 geometric data types

• point:p ˜ – represents place • vector: ~v – represents motion/offset/free arrow between 2 ordered points • coordinate vector: c • coordinate system ~st – “basis” for vectors – “frame” is for points

1 vectors vs coordinate vectors

• a point is a geometric entity in a 3D Euclidean space. – We will not present a formal mathematical definition of a point (This is a somewhat deep question) What is important is how points and lines in space “behave”. Not what they “are”. • a vector is a geometric entity, (motion/offset/free arrow between points), in a 3D Euclidean space • a coordinate vector is a set of numbers used to specify a vector given an agreed coordinate system vector space

• a set • has an addition operation • has a scalar multiplication with reals. • some other “special properies” – addition is associative and commutative – scalar mul must distribute across vector add

α(~v + ~w) = α~v + α ~w example vector spaces

• R3 (triplets of numbers) • quadratic functions, such as 5x2 + 2x − 7. – they can be added and scalar multiplied appropriately • this class: free-arrow between two ordered points in a euclidean space. – they can be added and scalar multiplied appropriately – these “vectors” should not be thought of as numerical objects. coordinate system: basis

• a basis is a minimal set of vectors that we can use to get to all of the vectors using our ops. – minimal == linearly independent • dimension is number of basis elements needed • for us it will be 3 • so basis can be used to address all of the vectors uniquely using coordinates

X ~ ~v = cibi i

• so a basis (a non-numerical object) can be used to represent a vector (non-numerical) with a set of coordinates (a numerical object!). shorthand

• write this as   c1 X ~ h ~ ~ ~ i ~v = cibi = b1 b2 b3  c2  i c3

• rhs uses the “two finger rule” from linear algebra, but the experssion is mixed numeric/geometric.

2 – we will use two finger rule alot in what follows. • even shorter

~v = ~btc linear transformation

• a linear tform L maps from V to V • satisfies 2 rules

L(~v + ~u) = L(~v) + L(~u) L( ~αv) = αL(~v)

• we will use the notation ~v → L(~v) – book uses ⇒ linear tforms and matrices

• linear transformation can be exactly specified by telling us its effect on the basis vectors. • linear transforms can be expressed with matrix multiplication • Linearity implies

X ~ X ~ ~v → L(~v) = L( cibi) = ciL(bi) i i

• in our shorthand this is     c1 c1 h ~ ~ ~ i h ~ ~ ~ i b1 b2 b3  c2  → L(b1) L(b2) L(b3)  c2  c3 c3

~ • each L(bi) can ultimately be written as some linear combination of the original basis vectors using numbers Mi,j

h ~ ~ ~ i L(b1) L(b2) L(b3) =   M1,1 M1,2 M1,3 h ~ ~ ~ i b1 b2 b3  M2,1 M2,2 M2,3  M3,1 M3,2 M3,3 so

• a linear transform operating on a vector can be expressed as   c1 h ~ ~ ~ i b1 b2 b3  c2  c3

    M1,1 M1,2 M1,3 c1 h ~ ~ ~ i → b1 b2 b3  M2,1 M2,2 M2,3   c2  M3,1 M3,2 M3,3 c3 well defined ops

• vector to vector

~btc → ~btMc

– see fig

3 • basis to basis,

h ~ ~ ~ i h ~ ~ ~ i b1 b2 b3 → L(b1) L(b2) L(b3)

• so this can be written as

~bt → ~btM

– see fig • coordinate vector to coordinate vector using two-finger rule. (this is the one we will see in code, but not until then).

c → Mc identity and inverse

• the identity matrix I implements to “do nothing” transform • an inverse matrix has the property MM −1 = M −1M = I • not every matrix has an inverse, but nice ones do, and all of our matrices are nice.

matrices for change of basis

• we just saw as an intermediate result an expression of the form   ~a1 ~a2 ~a3 =   M1,1 M1,2 M1,3 h ~ ~ ~ i b1 b2 b3  M2,1 M2,2 M2,3  M3,1 M3,2 M3,3

• or in shorthand

~at = ~btM ~atM −1 = ~bt

• this is not a transformation, but a statement of equality. • we have used a matrix to express one named basis with respect to another. • this will be useful too. • we can also use this to have different expressions for the same vector

~v = ~btc = ~atM −1c

• ex 2.1 and 2.2

Measurements

• we have some measurement operations that take in geometric objects and output numbers. • in a Euclidean space, we can measure the distances between point pairs, and the length k ~v k of a vector ~v. • in a Euclidean space, we can measure the angle θ made by two non-zero vectors ~v and ~w. • Given these, we can also define the (less intuitive) notion of a dot prodcut between two vectors:

~v · ~w = cos(θ) k ~v kk ~w k

• there are a bunch of “special properties” that this dot posseses, which we won’t go into. dot

4 • we can actually go the other way around. • once we have the dot product, we can use this to define length and angle.

k ~v k2:= ~v · ~v

~v · ~w cos(θ) := k ~v kk ~w k

• so in math we often think of the dot product as the core measurement. • indeed in math, one is allowed to come up with a new dot product as long as it satisfies the “special properties”, and then use that to come up with a new notion of length and angle. ortho

• 2 vectors are orthogonal if ~v · ~w = 0. • orthonormal basis • right handed (cyclically-ordered) basis • dot product in orthonormal basis

~ t ~ t X ~ X ~ b c · b d = ( cibi) · ( djbj) i j X X ~ ~ = cidj(bi · bj) i j X = cidi i

• so given an orthornormal coordinate system, we now have a nice numerical way to compute the dot product! – two cooridinate vectors in, one number out cross

• in 3d Euclidean space we also have a cross product on two ordered vectors. • the output is the vector

~v × ~w :=k v k k w k sin(θ) ~n

• where ~n is the right handed normal of {~v, ~w}

• in a r.h. o.n. basis, the coordinates of (~btc) × (~btd) are   c2d3 − c3d2  c3d1 − c1d3  c1d2 − c2d1 rotations

• transform that preserves dot product between vector pairs • preserves right handedness between ordered vector triples • so maps r.h.o.n. basis to another • in 3d, every rotation fixes an axis, and rotates some angles r.h. about that axis. comments

• rotations about different axes do not commute

5 • composition of two rots about two axes is a rotation about some third axis. – though the identiy of this third axis is not obvious.

2D rotations

• rotate by θ degrees counter clockwise about the origin

h i  x  ~b ~b 1 2 y h i  cos θ − sin θ   x  → ~b ~b 1 2 sin θ cos θ y

• and we can rotate the basis as h ~ ~ i b1 b2 h i  cos θ − sin θ  → ~b ~b 1 2 sin θ cos θ

3d rotations

• rotate a point by θ degrees around the z axis of the basis

 x  h ~ ~ ~ i b1 b2 b3  y  z  c −s 0   x  h ~ ~ ~ i → b1 b2 b3  s c 0   y  0 0 1 z

• where c ≡ cosθ, and s ≡ sinθ. • fixes points on z axis • for points in z = k plane, it is like a 2D rotation • basis is important (z direction) more 3d rotations

• around x axis  x  h ~ ~ ~ i b1 b2 b3  y  z  1 0 0   x  h ~ ~ ~ i → b1 b2 b3  0 c −s   y  0 s c z

• forward rotation around the y axis

 c 0 s   0 1 0  −s 0 c arbitrary rotation

• can get any rotation by applying one x,y,z • can get any rotation by applying one x,y,x – called Euler angles

6 – visualize with set of gimbals

t • one can specify rotation with unit vector axis kˆ := [kx, ky, kz] and θ using matrix

 2  kxv + c kxkyv − kzs kxkzv + kys 2  kykxv + kzs kyv + c kykzv − kxs  2 kzkxv − kys kzkyv + kxs kz v + c

• where v ≡ 1 − c other linear transforms

• uniform scales (common)

 a 0 0   0 a 0  0 0 a

• non-uniform scales (used for modeling)   ax 0 0  0 ay 0  0 0 az

• shears (rare)

 1 b 0   0 1 0  0 0 1

• ex 2.3, 2.4

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