Objectives. Geometry. Coordinate-Free Geometry. Basic Elements. Transformations to Change Coordinate Systems. Scalars

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1 Objecties Geometry CS Interactie Computer Graphics Prof. Daid E. Breen Department of Computer Science Introduce the elements of geometry - Scalars - Vectors - Points Deelop mathematical operations among them in a coordinate-free manner Define basic primities - Line segments - Polygons Basic Elements Coordinate-Free Geometry Geometry is the study of the relationships among objects in an n-dimensional space - In computer graphics, we are interested in objects that exist in three dimensions Want a minimum set of primities from which we can build more sophisticated objects We will need three basic elements - Scalars - Vectors - Points When we learned simple geometry, most of us started with a Cartesian approach - Points were at locations in space p=(x,y,z) - We deried results by algebraic manipulations inoling these coordinates This approach was nonphysical - Physically, points exist regardless of the location of an arbitrary coordinate system - Most geometric results are independent of the coordinate system - Example Euclidean geometry: two triangles are identical if two corresponding sides and the angle between them are identical Transformations to Change Coordinate Systems Scalars coordinate systems point P M M M = T (,) = T (,) S(.,.) = T (.,.) R( ) Need three basic elements in geometry - Scalars, Vectors, Points Scalars can be defined as members of sets which can be combined by two operations (addition and multiplication) obeying some fundamental axioms (associatiity, commutiity, inerses) Examples include the real and complex number systems under the ordinary rules with which we are familiar Scalars alone hae no geometric properties 99 Foley/VanDam/Finer/Huges/Phillips ICG

2 Vectors Vector Operations Physical definition: a ector is a quantity with two attributes - Direction - Magnitude Examples include - Force - Velocity - Directed line segments Most important example for graphics Can map to other types Eery ector has an inerse - Same magnitude but points in opposite direction Eery ector can be multiplied by a scalar There is a zero ector - Zero magnitude, undefined orientation The sum of any two ectors is a ector - Use head-to-tail axiom - α u w Linear Vector Spaces Vectors Lack Position Mathematical system for manipulating ectors Operations - Scalar-ector multiplication u=α - Vector-ector addition: w=u+ Expressions such as =u+w-r Make sense in a ector space These ectors are identical - Same length and magnitude Vectors spaces insufficient for geometry - Need points 9 Points Affine Spaces Location in space Operations allowed between points and ectors - Point-point subtraction yields a ector - Equialent to point-ector addition =P-Q P=+Q Point + a ector space Operations - Vector-ector addition - Scalar-ector multiplication - Point-ector addition - Scalar-scalar operations For any point define - P = P - P = (zero ector)

3 Lines Parametric Form Consider all points of the form - P(α)=P + α d - Set of all points that pass through P in the direction of the ector d This form is known as the parametric form of the line - More robust and general than other forms - Extends to cures and surfaces Two-dimensional forms - Explicit: y = mx +h - Implicit: ax + by +c = - Parametric: x(α) = αx + (-α)x y(α) = αy + (-α)y Rays and Line Segments Conexity If α >=, then P(α) is the ray leaing P in the direction d If we use two points to define, then P( α) = Q + α (R-Q) = Q+α = αr + (-α)q For <= α < = we get all the points on the line segment joining R and Q An object is conex iff for any two points in the object all points on the line segment between these points are also in the object Q P conex P Q not conex Affine Sums Conex Hull Consider the sum P=α P +α P +..+α n P n If α +α +..α n = in which case we hae the affine sum of the points P,P,..P n If, in addition, α i >=, we hae the conex hull of P,P,..P n Smallest conex object containing P,P,..P n Formed by shrink wrapping points

4 Cures and Surfaces Cures are one parameter entities of the form P(α) where the function is nonlinear Surfaces are formed from two-parameter functions P(α, β) - Linear functions gie planes and polygons Planes A plane can be defined by a point and two ectors or by three points P P(α) P(α, β) R u P(α,β)=R+αu+β R Q P(α,β)=R+α(Q-R)+β(P-R) 9 Triangles Barycentric Coordinates conex sum of P and Q for <=α,β<=, we get all points in triangle conex sum of S(α) and R Triangle is conex so any point inside can be represented as an affine sum P(α, α, α )=α P+α Q+α R where α +α +α = α i >= The representation is called the barycentric coordinate representation of P Normals Eery plane has a ector n normal (perpendicular, orthogonal) to it From point-two ector form P(α,β)=R+αu+β, we know we can use the cross product to find n = u and the equialent form (P(α)-P) n= Representation P u

5 Objecties Linear Independence Introduce concepts such as dimension and basis Introduce coordinate systems for representing ectors spaces and frames for representing affine spaces Discuss change of frames and bases Introduce homogeneous coordinates A set of ectors,,, n is linearly independent if α +α +.. α n n = iff α =α = = If a set of ectors is linearly independent, we cannot represent one in terms of the others If a set of ectors is linearly dependent, at least one can be written in terms of the others Dimension Representation In a ector space, the maximum number of linearly independent ectors is fixed and is called the dimension of the space In an n-dimensional space, any set of n linearly independent ectors form a basis for the space Gien a basis,,., n, any ector can be written as =α + α +.+α n n where the {α i } are unique Until now we hae been able to work with geometric entities without using any frame of reference, such as a coordinate system Need a frame of reference to relate points and objects to our physical world. - For example, where is a point? Can t answer without a reference system - World coordinates - Camera coordinates Coordinate Systems Example Consider a basis,,., n A ector is written =α + α +.+α n n The list of scalars {α, α,. α n }is the representation of with respect to the gien basis We can write the representation as a row or column array of scalars α a=[α α. α n ] T = α. α n 9 = + - a=[,, ] T Note that this representation is with respect to a particular basis For example, in OpenGL we start by representing ectors using the object basis but later the system needs a representation in terms of the camera or eye basis

6 Coordinate Systems Frames Which is correct? A coordinate system is insufficient to represent points If we work in an affine space we can add a single point, the origin, to the basis ectors to form a frame Both are because ectors hae no fixed location P Representation in a Frame Confusing Points and Vectors Frame determined by (P,,, ) Within this frame, eery ector can be written as =α + α +.+α n n Eery point can be written as P = P + β + β +.+β n n Consider the point and the ector P = P + β + β +.+β n n =α + α +.+α n n They appear to hae the similar representations p=[β β β ] =[α α α ] which confuses the point with the ector p A ector has no position Vector can be placed anywhere point: fixed A Single Representation Homogeneous Coordinates If we define P = and P =P then we can write =α + α +α = [α α α ] [ P ] T P = P + β + β +β = [β β β ] [ P ] T Thus we obtain the four-dimensional homogeneous coordinate representation = [α α α ] T p = [β β β ] T The homogeneous coordinates form for a three dimensional point [x y z] is gien as p =[x y z w] T =[wx wy wz w] T We return to a three dimensional point (for w ) by x x /w y y /w z z /w If w=, the representation is that of a ector Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions For w=, the representation of a point is [x y z ]

7 Homogeneous Coordinates and Computer Graphics Change of Coordinate Systems Homogeneous coordinates are key to all computer graphics systems - All standard transformations (rotation, translation, scaling) can be implemented with matrix multiplications using x matrices - Hardware pipeline works with dimensional representations - For orthographic iewing, we can maintain w= for ectors and w= for points - For perspectie we need a perspectie diision Consider two representations of the same ector with respect to two different bases. The representations are a=[α α α ] b=[β β β ] where =α + α +α = [α α α ] [ ] T =β u + β u +β u = [β β β ] [u u u ] T Representing second basis in terms of first Matrix Form Each of the basis ectors, u,u, u, are ectors that can be represented in terms of the first basis The coefficients define a x matrix M = u = + + u = + + u = + + and the bases can be related by a=m T b see text for numerical examples 9 Change of Frames Representing One Frame in Terms of the Other We can apply a similar process in homogeneous coordinates to the representations of both points and ectors u u Consider two frames: (P,,, ) Q (Q, u, u, u ) P u Any point or ector can be represented in either frame We can represent Q, u, u, u in terms of P,,, Extending what we did with change of bases u = + + u = + + u = + + Q = P defining a x matrix M =

8 Working with Representations The World and Camera Frames Within the two frames any point or ector has a representation of the same form a=[α α α α ] in the first frame b=[β β β β ] in the second frame where α = β = for points and α = β = for ectors and a=m T b The matrix M is x and specifies an affine transformation in homogeneous coordinates When we work with representations, we work with n-tuples or arrays of scalars Changes in frame are then defined by x matrices In OpenGL, the base frame that we start with is the world frame Eentually we represent entities in the camera frame by changing the world representation using the model-iew matrix Initially these frames are the same (M=I) Moing the Camera If objects are on both sides of z=, we must moe camera frame M = d