Describe the Orthographic and Perspective projections. How do we combine together transform matrices?


 Coral Hodge
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1 Aims and objectives By the end of the lecture you will be able to Work with multiple transform matrices Describe the viewing process in OpenGL Design and build a camera control APIs Describe the Orthographic and Perspective projections 1 Composing Matrices Composing Transforms How do we combine together transform matrices? We can multiply two matrices together to get a new transform M = RT Composing Transforms Order matters The effect of transforming a matrix by M is: Mv = RTv = R(Tv) This is the same as doing T then doing R TR does R then doing T Note that this is the opposite order from how it is written 2 GL Transform Matrices Transform Matrices in OpenGL OpenGL keeps a current transform matrix This is multiplied with all vertices that are drawn call it C Transform Matrices in OpenGL You can all glloadidentity to set C to the identity matrix C = I 1
2 Transform Matrices in OpenGL When you call glscale, glrotate and gltranslate the current matrix is multiplied by a scale, rotation or translation matrix C = CR Transform Matrices in OpenGL g l. g l L o a d I d e n t i t y ( ) ; g l. g l T r a n s l a t e f ( 1 0 0, 0, 0 ) ; g l. g l R o t a t e f ( 4 5, 0, 1, 0 ) ; g l. g l S c a l e ( 1. 5, 1. 5, 1. 5 ) ; If T is a translate, R a rotate and S a scale this code does: C = ITRS NB the transforms are done in the opposite order from how they are written in the code Transform Matrices in OpenGL g l. g l T r a n s l a t e f ( 1 0 0, 0, 0 ) ; box ( ) ; g l. g l T r a n s l a t e f ( 2 0 0, 0, 0 ) ; box ( ) ; / / T1 / / T2 What does this code do? C = CT1 draw a box C = CT2 draw a box When the 2nd box is drawn, C = T1T2 Transform Matrices in OpenGL We want to be able to draw different objects in different positions This means creating a transform for one object and drawing it then getting rid of that transform before drawing the next object We use a method called glpushmatrix 2
3 Matrix Stack Matrices in OpenGL are stored as a stack A stack is like a stack of paper Matrix Stack 3
4 Matrices in OpenGL are stored as a stack A stack is like a stack of paper You can add things to the top of a stack (Push) Matrix Stack 4
5 Matrices in OpenGL are stored as a stack A stack is like a stack of paper Or remove them again from the top (Pop) Matrix Stack 5
6 Matrices in OpenGL are stored as a stack A stack is like a stack of paper Or remove them again from the top (Pop) Matrix Stack The last thing to be put on is the first to be taken off Last in, first out Matrix Stack When you add a matrix to the stack it is multiplied into the current matrix When remove it, it is removed from the current matrix 6
7 Matrix Stack Storing matrices as a stack means that the most recent transforms are the ones you remove first Generally what you want Global transforms that affect the whole scene are at the bottom Transforms that only affect a single object at the top PushMatrix and PopMatrix glpushmatrix() creates a new matrix and puts it on the top of the stack You can then do any transforms you like glpopmatrix() will then remove the matrix from the stack i.e. it will remove all the transforms you have done PushMatrix and PopMatrix g l. g l P u s h M a t r i x ( ) ; g l. g l T r a n s l a t e f ( 1 0 0, 0, 0 ) ; box ( ) ; g l. g l P o p M a t r i x ( ) ; g l. g l P u s h M a t r i x ( ) ; g l. g l T r a n s l a t e f ( 2 0 0, 0, 0 ) ; box ( ) ; g l. g l P o p M a t r i x ( ) ; / / T1 / / T2 For multiple objects: glpushmatrix() before drawing each object Do all the transforms for that object glpopmatrix() to get rid of the transforms before moving on to the next matrix 3 Coordinate System Transforms Coordinate Systems 7
8 The world in 3D geometry is defined by: The position of the origin The direction of x, y and z (called the axes) Together these are called a coordinate system Often it is convenient to use different coordinate system definitions in different situations Coordinate Systems 8
9 Objects are positioned relative to the cooridinate system Coordinate Systems 9
10 But we can define a separate coordinate system for an object This can be very useful for modelling an object e.g. the origin can be in the middle of a sphere e.g. The axes can run along the walls of a house Coordinate Systems 10
11 By transforming an object we are changing the coordinate system of an object relative to the world Translating the origin Rotating the axes Coordinate Systems 11
12 If you have lots of axes you will have lots of coordinate systems Coordinate Systems 12
13 Sometimes you want to share a coordinate system between objects If you have a sofa and a chair in your house you want to position them relative to the house not the world You can have a heirarchy of coordinate systems sofa and chair have a CS relative to the CS of the house which is relative to the world CS Coordinate Systems 13
14 If you rotate the coordinate system of the house, the furniture moves with it You don t have to move items individually 4 Viewing and Projection Aims and objectives By the end of the lecture you will be able to Work with multiple transform matrices Describe the viewing process in OpenGL Design and build a camera control APIs Describe the Orthographic and Perspective projections Viewing and Projection Up to now we have talked about creating an moving objects 14
15 But how do we draw them on screen A complex process Position a virtual camera Project the 3D object onto a flat screen Draw the individual polygons onto the pixels of the screen Will talk about the first two 5 Model, View and Projection Matrix Viewing Coordinate Systems Each object has its own coordinate system This is converted into a world coordinate system To view it we have to convert it into another coordinate system: The coordinate system of the camera This means that we are viewing everything relative to the camera Camera Coordinate System 15
16 Y Z X camera has its own coordinate system The The origin is at the position of the camera z points out of the camera y points up x points left (or sometimes right) Viewing Matrices You need 3 matrices to view an object The Modelling matrix positions an object relative to the world The Viewing matrix transforms it into the coordinates of the camera The Projection matrix projects it onto a flat screen OpenGL combines the first 2 into a single ModelView matrix 16
17 Viewing Matrices To change one of these matrices use the method gl.glmatrixmode gl.glmatrixmode(gl.gl_modelview) gl.glmatrixmode(gl.gl_projection) You can then use normal transform commands or the projection matrix commands I ll talk about later 6 Camera Control Camera control The point of all this is to be able to control the camera You put transforms into the ModelView matrix Traslations position the camera Rotations change where its pointing Camera control gl. glmatrixmode ( gl.gl_modelview ) ; g l. g l L o a d I d e n t i t y ( ) ; g l. g l T r a n s l a t e f ( 0. 0 f, 0. 0 f, f ) ; Move the camera back along the z axis This can work if you model everything at the origin Camera control gl. glmatrixmode ( gl.gl_modelview ) ; g l. g l L o a d I d e n t i t y ( ) ; g l. g l R o t a t e f ( c a m e r a R o t a t i o n. x, 1, 0, 0 ) ; g l. g l R o t a t e f ( c a m e r a R o t a t i o n. y, 0, 1, 0 ) ; g l. g l T r a n s l a t e f ( 0. 0 f, 0. 0 f, f ) ; Rotate the camera as well The camerarotation variable can be set by mouse rotation glulookat p g l. g l u. glulookat ( eyex, eyey, eyez, atx, aty, atz, upx, upy, upz ) ; A handy function that is part of the glu library (OpenGL utilities) Sets the camera position based on 3 vectors eye: the position of the camera at: where it is looking at up: the up direction 17
18 glulookat p g l. g l u. glulookat ( eyex, eyey, eyez, atx, aty, atz, upx, upy, upz ) ; glulookat gl. glmatrixmode ( gl.gl_modelview ) ; g l. g l L o a d I d e n t i t y ( ) ; p g l. g l u. glulookat ( 0, 0, 500, 200, 0, 0, 0, 1, 1 ) ; Put the camera at 500 z look at an object at 200 x you will normally use y as up Moving cameras glulookat is good for setting up a static camera most of the time we want to be able to move it around navigate based on user input glulookat can get complex use glulookat to set up a sensible starting point use transforms to move around 18
19 Model View Matrix The model view matrix includes both the position of the model and the camera Any transform in there can be interpreted as either 1. keeping world still and moving the camera 2. keeping the camera still and moving the world While 2 might sound strange it can often be easier to think about Transforms are hard to think about, any little bit helps Object viewing interface If you have an application about viewing an object a good interface is to be able to: Spin the camera around the object Zoom in and out Use 2 rotations to spin about x and y The transform by z to zoom gl. glmatrixmode ( gl.gl_modelview ) ; g l. g l L o a d I d e n t i t y ( ) ; p g l. g l u. glulookat ( 0. 0 f, 0. 0 f, 1.0f, 0. 0 f, 0. 0 f, 0. 0 f, 0. 0 f, 1. 0 f, 0. 0 f ) ; g l. g l T r a n s l a t e f ( 0. 0 f, 0. 0 f, zoom ) ; g l. g l R o t a t e f ( c a m e r a R o t a t i o n. x, 1, 0, 0 ) ; g l. g l R o t a t e f ( c a m e r a R o t a t i o n. y, 0, 1, 0 ) ; Object viewing interface We are keeping the camera still and rotating the world To do the opposite we reverse the order of the translation and rotations More like a first person camera First Person Camera For a proper first person shooter type camera we need to be able to: 1. Rotate the camera to change where you are looking 2. Move in the direction you are looking gl. glmatrixmode ( gl.gl_modelview ) ; g l. g l L o a d I d e n t i t y ( ) ; p g l. g l u. glulookat ( 0. 0 f, 0. 0 f, 1.0f, 0. 0 f, 0. 0 f, 0. 0 f, 0. 0 f, 1. 0 f, 0. 0 f ) ; g l. g l R o t a t e f ( c a m e r a R o t a t i o n. x, 1, 0, 0 ) ; g l. g l R o t a t e f ( c a m e r a R o t a t i o n. y, 0, 1, 0 ) ; g l. g l T r a n s l a t e f ( foward. x, f o r w a r d. y, f o r w a r d. z ) ; 19
20 First Person Camera We need to move in the current forward direction (based on camera rotation) Take the zaxis and rotate it by the current camera rotation GL doesn t have a method to translate by a matrix do it by hand based on the trigonometry equations (see the code for more details) Third Person Camera Move your avatar (character, vehicle) keep the camera behind the avatar and pointing at it no longer directly controlling the camera 7 Projection and clipping Projection We need to take a 3D scene and represent it as a 2D image This is a fundamental change of representation loses information Different ways of doing this with different mathematical representations Projection 20
21 Orthographic Projection Orthographic Projection Points are projected in a parallel line to the screen Preserves distances Useful for exact measurement and scientific work Not very realistic Perspective Projection 21
22 Perspective Projection Perspective Projection Models how points in the distance grow smaller lines are drawn from the obejct to the viewers eye Produces realistic images for normal viewing Perspective Projection 22
23 Clipping Projection onto 2D means that there is a lot of the scene that we don t see To make the process more efficient, we avoid drawing objects that won t get projected Called clipping (different from objects that are hidden behind other objects) Clipping 23
24 Clipping Clipping is controlled by 6 planes: near, far left, right top, bottom We can use these to define the parameters of our project glortho g l. g l O r t h o ( GLdouble l e f t, GLdouble r i g h t, GLdouble bottom, GLdouble top, GLdouble near, GLdouble far ) ; Does orthonormal projection and clipping Creates a Projection Matrix (won t go into the details) Perspective Clipping 24
25 glfrustum g l. glfrustum ( GLdouble l e f t, GLdouble r i g h t, GLdouble bottom, GLdouble top, GLdouble near, GLdouble far ) ; Does perspective projection and clipping Pretty much the same as glortho top, bottom, left and right are defined at the near clipping plane 25
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