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Lecture 3: Transformations Todays lecture • Object representations in CG • Review of linear algebra • Linear transformations (rotation, scaling) with matrices. • Translation and homogeneous coordinates. • Main point: A lot of interesting transforms can be expressed with matrix multiplication Todays lecture in context Classical graphical rendering pipeline 3D Object representation for CG • Surface representations • Discrete explicit surfaces (points, polygons) • Explicit functional surface definitions (x,y,z) <- f(a,b,…) • Implicit surface definitions f(x,y,z) = C • Solid (volumetric) models • Discrete volumetric models (voxels) • Implicit solid models f(x,y,z) in range [ ] 3D Object representation for CG • An object is described by its surface (i.e. objects are hollow). • An object is specified by a set of 3D points (vertices). • An object is composed of (or can be approximated by) flat convex polygons. • Transform an object by transforming its vertices Smooth objects? • May be approximated by polygons • If the (screen)size of each polygon is smaller than the size of a pixel, there is no visual difference. • Modern graphics cards can render LOTS of triangles very fast. • We will talk more about smooth surfaces later in the course. CG-dinosaur in Jurrasic Park Other object representations Solid/Volumetric modelling Objects are composed of a set of Voxels = 3D pixels Volumetric Modelling Volumetric rendering of a foot. (Project at the end of the course) Other object representations Implicit surfaces, F(x,y,z)=C y x r=1 Sphere defined implicitly by 𝑥 2 + 𝑦 2 + 𝑧 2 = 12 Scalars points and vectors • Scalars – Real (or complex) numbers • Points – Locations in space (no size or shape) • Vectors – Directions in space (magnitude but no position) 1.2 -2.3 7.3i Vector operations Review of vector operations: • Adding/Subtracting two vectors • Multiplication by scalars • Magnitude of a vector • Normalizing a vector Vector Addition a = (2, 4) b = (2, 2) y x Vector Addition a = (2, 4) b = (2, 2) a + b = c = (4, 6) y x Vector Addition a = (2, 4) b = (2, 2) b + a = c = (4, 6) y x Vector Subtraction a = (2, 4) b = (2, 2) a - b = (2, 4) – (2, 2) = (0, 2) = d y x Vector Subtraction y a = (2, 4) b = (2, 2) b - a = (2, 2) - (2, 4) = (0, -2) = e ≠ d x Order is important when subtracting vectors! Vector Multiplied by Scalar a = (4, 2) y x Vector Multiplied by Scalar b = 2 * a = 2 * (4, 2) = (8, 4) y x Vector Multiplied by Scalar b = -0.5 * a = -0.5 * (4, 2) = (-2, -1) y x Length of Vector a = (ax, ay) ||a|| = 𝑎𝑥 2 + 𝑎𝑦 2 Length of vector a (Euclidian norm) (Remember: Pythagoras' Theorem) Length of Vector a = (ax, ay) ||a|| = 𝑎𝑥 2 + 𝑎𝑦 2 Length of vector a (Euclidian norm) b = (bx, by, bz) ||b|| = 𝑏𝑥 2 + 𝑏𝑦 2 + 𝑏𝑧 2 Length of Vector a = (ax, ay) ||a|| = 𝑎𝑥 2 + 𝑎𝑦 2 Length of vector a (Euclidian norm) b = (bx, by, bz) ||b|| = 𝑏𝑥 2 + 𝑏𝑦 2 + 𝑏𝑧 2 ||a|| = 𝒂•𝒂 Dot product Normalizing a Vector Normalize a vector = set its length to 1. Divide the vector by its Euclidian norm (length) Skriv en ekvation här. 𝒂 𝒂= |𝒂| Matrix multiplication 𝑎 𝑑 𝑔 𝑏 𝑒 ℎ 𝑐 𝑓 𝑖 𝑥 𝑦 𝑧 = 𝑎 𝑑 𝑔 𝑏 𝑒 ℎ 𝑐 𝑓 𝑖 𝑗 𝑚 𝑝 𝑘 𝑛 𝑞 𝑎𝑗 + 𝑏𝑚 + 𝑐𝑝 𝑑𝑗 + 𝑒𝑚 + 𝑓𝑝 𝑔𝑗 + ℎ𝑚 + 𝑖𝑝 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 𝑑𝑥 + 𝑒𝑦 + 𝑓𝑧 𝑔𝑥 + ℎ𝑦 + 𝑖𝑧 𝑙 𝑜 = 𝑟 𝑎𝑘 + 𝑏𝑛 + 𝑐𝑞 𝑑𝑘 + 𝑒𝑛 + 𝑓𝑞 𝑔𝑘 + ℎ𝑛 + 𝑖𝑞 𝑎𝑙 + 𝑏𝑜 + 𝑐𝑟 𝑑𝑙 + 𝑒𝑜 + 𝑓𝑟 𝑔𝑙 + ℎ𝑜 + 𝑖𝑟 Linear combinations of vectors • Convex combinations (coefficients are all positive and add up to 1) • E.g., linear interpolation: p(t)=(t)a +(1-t)b where a and b are points and 0≤t≤1. Linear combinations of vectors a = (2, 6) b = (10, 4) c = t*b + (1-t)*a y t = 0.0 t = 0.25 t = 0.50 t = 0.75 t = 1.0 x Linear combinations of vectors What are convex combinations used for in CG? Can you imagine some examples? Linear combinations of vectors Morphingthe“Ballephant” Source object 100% Elephant Target object 50% Elephant 50% Ball 25% Elephant 75% Ball 100% Ball Intermediate objects are linear combinations of corresponding vertices in the source and target geometries module2s.rfx AGP2010 Dot products • Vectors are orthogonal if dot product is 0. • Vectors are less than 90o apart if dot product is positive. a•𝐛 = 0 a•𝐛 > 0 a•𝐛 < 0 Dot products •Perpendicular projection of a onto b is (a•𝐛)𝐛 Dot products Geometric interpretation in CG: Perpendicular projection of a onto b is (a•𝐛)𝐛 a b 𝐛 c c = projection of a onto b Dot products What can the dot product be used for in CG? Can you imagine some applications? Dot products What can the dot product be used for in CG? Illumination/shading (angle between surface and light direction) Backface culling (angle between viewing direction and surface) Collision detection (distance point / plane) Reflection of light rays (mirror) or object motion (bouncing) … Dot products • How do we get the reflection b of a vector a? a b Dot products • Reflection b of a is –a+2(a•𝒏)𝒏 where 𝒏 is the unit normal vector to the reflection surface. a 𝒏 -a b (a•𝒏)𝒏 (a•𝒏)𝒏 Cross products Cross products Determinant Right hand rule Cross products Geometric interpretation and usage of the cross product in CG: ◦ a ₓ b is a vector perpendicular to a and b ◦ Length of this vector is equivalent with the area of the parallelogram defined by a and b ◦ Estimating surface normal vectors for L&S ◦ Determining normal basis / reference frames ◦ (Test for collinearity of vectors), but dot product is faster ◦ (Angle between vectors) , but dot product is faster Transformations • To create and move objects we need to be able to transform objects in different ways. • There are many classes of transformations. – Translation (move around) – Rotation – Scaling • Can you think of more? Transformations • To create and move objects we need to be able to transform objects in different ways. • There are many classes of transformations. – Translation (move around) – Rotation – Scaling • Can you think of more? - Shear - Mirroring/Flipping Translation Simply add a translation vector Scaling (about the origin) Multiply by a scale factor Rotation (about the origin) P(x,y) in polar coordinates P'(x',y') in polar coordinates Rotation (about the origin) P(x,y) in polar coordinates P'(x',y') in polar coordinates Substitute x and y Vectors and matrices (2D) Rotation on matrix form: (Rotation around the origin.) Arbitrary rotation center (0,0) Arbitrary rotation center •Translate the rotation center to the origin •Rotate •Translate back Vectors and matrices (2D) Translation Rotation Scaling Vectors and matrices (2D) Translation x’andy’resultfroman additional independent factor Rotation Scaling x’andy’arelinear combinations of x, and y Homogenous coordinates “Steppinguponedimension”: If W=1, this is called a Homogenous coordinate Note: These are 2D homogenous coordinates and not 3D (spatial) coordinates. Homogenouscoordinatesarea“mathematicalvehicle”tomodel various spatial transformations of coordinates in a uniform way. Example 1 0 0 1 0 0 −2 3 1 4 6 1 = 2 9 1 This way, matrix multiplication can be used also for translation! Why bother? • Compact notation for concatenated transforms. • Translation followed by Rotation ≠ Rotation followed by Translation Using homogenous coordinates Translation Rotation Scaling What about the last element W? • If W=0, then P is not affected by translations • Be aware of this during the assignments! Concatenation of transforms The order of matrices (right to left) is important! Example of concatenated transforms: 1 P′ = 0 0 0 1 0 −4 1 0 −2 ∙ 0 2 1 0 0 Translation 0 0 −1 0 ∙ 1 0 1 0 0 Scaling 0 1 0 0 ∙ 0 1 1 0 0 Rotation 𝑥 4 2 ∙ 𝑦 1 1 Translation P Moving to 3D • Translation and scaling are the same • Rotation is a bit more complicated, and becomes Rx, Ry and Rz. • Observe that RxRy ≠RyRx 3D transformations Translation Scaling 3D transformation 3 X Rotation Y Rotation Z Rotation Arbitrary rotation Rotation about an arbitrary axis a: • Find a matrix M that aligns e.g. (1,0,0) with a. • Apply M-1. • Rotate around (1,0,0) • Apply M. How do we find M? • The matrix M represents a change of coordinate system. • This (among other things) is the topic of the next lecture. Lab 2 • Transformations • • • Translation Rotation Scaling • Positioning of a camera in a graphical scene
