CGT520 Transformations

Transcription

1 CGT520 Transformations Bedrich Benes, Ph.D. Purdue University Department of Computer Graphics GLM A concise library for transforms and OpenGL mathematics All overloaded, predefined, optimized glm (OpenGL Mathematics) Download: It is header only It is GLSL friendly GLM Suggestions: Do not use namespace glm causes a lot of clashes use glm::func instead Use precompiled headers, it uses a lot of templates and can become slow to compile GLM - Example #include <glm/glm.hpp> #include <glm/gtc/type_ptr.hpp> #include <glm/gtc/matrix_transform.hpp> glm::vec4 TrMul(glm::vec4 p, glm::mat4 m) const glm::vec4 t=glm::vec4(1.0); return glm::vec4(m*p+t);

2 GLM OpenGL transformations Everything is done via shaders we will see more during this lecture OpenGL transformations Represented using matrices The matrix is multiplied from the right m11 m12 m13 m14 m21 m22 m23 m24 m31 m32 m33 m34 m41 m42 m43 m44 OpenGL transformations Each stage of the transformation pipeline can be represented by a different matrix They can be stored into a premultiplied matrix as a single ModelViewProjection or separated into three matrices Modeling, Viewing, and Projection

3 OpenGL transformations The vertex shader must perform a vertex transformation using the fixed order Transforms are defined in the same order GLSL Vertex Shader in vec4 iposition; uniform mat4 model; uniform mat4 view; uniform mat4 proj; void main() gl_position=proj*view*model*iposition; GLSL Vertex Shader in vec4 iposition; uniform mat4 model; uniform mat4 view; uniform mat4 proj; void main() gl_position=proj*view*model*iposition; Modeling Transforms Affect the objects in the world space Most commonly it is a linear transform We do not need to transform each element, but only the result.

4 Modeling Transforms Translate Translate Translates by a vector Scale Rotate around an arbitrary axis = x T y T z T Scale Scales by a vector Rotation Rotates by an angle around an axis = x s y s z s Source Wikipedia

5 GLM - implementation mat4 translate(mat4, vec3) mat4 scale (mat4, vec3) mat4 rotate (mat4,, vec3) If the mat4 is provided as the first argument, the result is multiplied and returned If mat4(1.0) is provided, it returns the transform matrix itself GLM - implementation So this is usually much faster m=glm::translate(m,glm::vec3(1,1,1)); m=glm::rotate (m,a,glm::vec3(0,0,1)); m=glm::translate(m,glm::vec3(1,1,1)); than this const glm::mat4 i(1.0); //identity matrix m=m*glm::translate(i,glm::vec3(1,1,1)); m=m*glm::rotate (i,a,glm::vec3(0,0,1)); m=m*glm::translate(i,glm::vec3(1,1,1)); GLM optimization-premultiply In the previous example we did: Modeling transforms the order of transformation is critical! But we can do m=glm::mat4(1.0); Display(); m=rotatef(m,45,0,0,1); m=translate(m,1,0,0); Display();

6 Modeling transforms the order of transformation is critical! Modeling transforms the order of transformation is critical because matrix multiplication is not commutative m=glm::mat4(1.0); Display(); m=translate(m,1,0,0); m=rotatef(m,45,0,0,1); Display(); C =RT C C C =TR Modeling Transforms Example The first approach: the grand fixed coordinate system we must read the transform in the inverse order 1: m=glm::mat4(1.0) 2: m=translatef(m,1,0,0); 3: m=rotatef(m,45,0,0,1); 4: DrawObject(); The second approach: local coordinate system tied to object operations are applied to the local coordinate system operations occur in natural order

7 Modeling Transform the second approach is more useful for modeling a sets of joints (robot arms) forward kinematics Modeling Transform - example the first approach is problematic in cases of scaling (non-uniform ~ axes may be non perpendicular) Modeling Transform - example scale translate 27 Modeling Transform - example //shapes[0] and shapes[1] are two cubes //the lower box first glm::mat4 m=glm::mat4(1.0); //reset m=glm::rotate(m,ftime,glm::vec3(0,0,1)); m=glm::translate(m,glm::vec3(0,0.5,0)); m=glm::scale(m,glm::vec3(0.1,1,0.1)); //set the matrix in the shader gluniformmatrix4fv(mp,1,gl_false, glm::value_ptr(m)); shapes[0]->render();

8 Modeling Transform - example //shapes[0] and shapes[1] are two cubes //the upper box then m=glm::scale(m,glm::vec3(10.f,1.0f,10.f)); m=glm::translate(m,glm::vec3(0,0.5,0)); m=glm::rotate(m,2*ftime,glm::vec3(0,0,1)); m=glm::translate(m,glm::vec3(0,0.5,0)); m=glm::scale(m,glm::vec3(0.1f,1.0f,0.1f)); //set the matrix in the shader gluniformmatrix4fv(mp,1,gl_false, glm::value_ptr(m)); shapes[1]->render(); GLSL Vertex Shader in vec4 iposition; uniform mat4 model; uniform mat4 view; uniform mat4 proj; void main() gl_position=proj*view*model*iposition; GLSL Vertex Shader in vec4 iposition; uniform mat4 model; uniform mat4 view; uniform mat4 proj; void main() gl_position=proj*view*model*iposition; Viewing Transform viewing transformation is rotating, translating, and scaling the camera viewing transformation should be issued before any modeling transformation in a program viewing transformation is issued first modeling transformation is discussed first viewing transformation are issued first i.e., first compose a scene then place a camera

9 lookat lookat(vec3 eye, vec3 target, vec3 upvector) invalid if Viewing Example Camera flying above a scene view=glm::lookat( vec3(r*sin(ftime),5,r*cos(ftime)),//eye vec3(0,0,0), //destination vec3(0,1,0)); //up //send it to the shader gluniformmatrix4fv(viewparameter,1, GL_FALSE,glm::value_ptr(view)); GLSL Vertex Shader in vec4 iposition; uniform mat4 model; uniform mat4 view; uniform mat4 proj; void main() gl_position=proj*view*model*iposition; GLSL Vertex Shader in vec4 iposition; uniform mat4 model; uniform mat4 view; uniform mat4 proj; void main() gl_position=proj*view*model*iposition;

10 Orthographic (Parallel) Projection mat4::ortho(left, right, bottom, top, near, far) Orthographic (Parallel) Projection Source Wikipedia It is in fact translation followed by scaling: Source Wikipedia Orthographic (Parallel) Projection Orthographic projection comments camera can be inside the volume ortho(-1,1,-1,1,-1,1) then also the object behind the camera are displayed Perspective Projection mat4::frustum(left, right, bottom, top, near, far) camera can be entirelly shifted ortho(-12,-10,-1,1,-1,1) it s like using a periscope

11 Perspective Projection Matrix Perspective Projection Frustum is not intuitive can be asymmetric Perspective Projection mat4::perspective(fovy, aspect, near, far) Perspective Projection gluperspective fovy is field of view in the y axis aspect is aspect ratio of the actual viewport it cannot be asymmetric (sometimes desired) fovy:

12 Perspective Projection perspective Could we set fovy from distance and size of an object? GLdouble FOVY(GLdouble size,gldouble dist) GLdouble theta; theta = 2.0*atan2(size/2.0, dist); return(180.0*theta/ ); Perspective Projection How to get image in double the maximal resolution? 1) Draw it four times setting the projections as depicted 2) Save the images 3) Glue them in Photoshop x=[-1, 0] y=[ 0, 1] x=[-1, 0] y=[-1, 0] x=[ 0, 1] y=[ 0, 1] x=[ 0, 1] y=[-1, 0] Viewport Viewport is the part of the window that is used for rendering void glviewport(glint x,glint y, GLsizei w, GLsizei h) - x and y specifies the lower left corner - w and h are size of the viewport rectangle viewport window Viewport Viewport is usually set in the reshape callback void Resize(int w, int h) glviewport(0,0,w,h); int main(int argc, char **argv) glutinit(&argc, argv); y w h glutdisplayfunc(display); glutreshapefunc(resize); x

13 Viewport Example - rendering into two viewports int x = 600, y = 300; gluperspective(60.0, x/(2.0*y), 1, 10.0);//!!! glviewport(0,0,x/2,y); RenderLeft(); glviewport(x/2,0,x/2,y); RenderRight() Reading GLSL Specification ngspec clean.pdf OpenGL Mathematics (GLM) Reading Edward Engel Interactive Computer Graphics: A Top-Down Approach with Shader-Based OpenGL (6 th Edition) Chapters 3 and 4 David Wolff, OpenGL 4.0 Shading Language Cookbook, PACKT Publishing 2011