TransformationsAdminRecap: OpenGL: Modeling TransformsRecap: OpenGL: Matrix ManipulationSlide 5OpenGL: Specifying ColorOpenGL: Specifying NormalsMore OpenGLTranslationsScalingSlide 11Slide 122-D RotationSlide 143-D RotationSlide 16Slide 17Slide 18Composing Canonical RotationsTransformationsCS 445/645Introduction to Computer GraphicsDavid Luebke, Spring 2003David Luebke 2 01/14/19Admin●Call roll●Assn 1■News flash: Assn 1 not necessarily a gimme■Turn-in instructions by email soon■Problems with lab?David Luebke 3 01/14/19Recap: OpenGL: Modeling Transforms●OpenGL provides several commands for performing modeling transforms:■glTranslate{fd}(x, y, z)○Creates a matrix T that transforms an object by translating (moving) it by the specified x, y, and z values■glRotate{fd}(angle, x, y, z)○Creates a matrix R that transforms an object by rotating it counterclockwise angle degrees about the vector {x, y, z}■glScale{fd}(x, y, z)○Creates a matrix S that scales an object by the specified factors in the x, y, and z directionsDavid Luebke 4 01/14/19Recap:OpenGL: Matrix Manipulation●Each of these postmultiplies the current matrix■E.g., if current matrix is C, then C=CS■The current matrix is either the modelview matrix or the projection matrix (also a texture matrix, won’t discuss)■Set these with glMatrixMode(), e.g.:glMatrixMode(GL_MODELVIEW);glMatrixMode(GL_PROJECTION);■WARNING: common mistake ahead!○Be sure that you are in GL_MODELVIEW mode before making modeling or viewing calls!○Ugly mistake because it can appear to work, at least for a while…David Luebke 5 01/14/19Recap:OpenGL: Matrix Manipulation●More matrix manipulation calls■To replace the current matrix with an identity matrix:glLoadIdentity()■Postmultiply the current matrix with an arbitrary matrix:glMultMatrix{fd}(float/double m[16])■ Copy the current matrix and push it onto a stack:glPushMatrix()■Discard the current matrix and replace it with whatever’s on top of the stack:glPopMatrix()■Note that there are matrix stacks for both modelview and projection modesDavid Luebke 6 01/14/19OpenGL: Specifying Color●Can specify other properties such as color■To produce a single aqua-colored triangle:glColor3f(0.1, 0.5, 1.0); glVertex3fv(v0); glVertex3fv(v1); glVertex3fv(v2);■To produce a Gouraud-shaded triangle:glColor3f(1, 0, 0); glVertex3fv(v0);glColor3f(0, 1, 0); glVertex3fv(v1);glColor3f(0, 0, 1); glVertex3fv(v2);■In OpenGL, colors can also have a fourth component (opacity)○Generally want = 1.0 (opaque);David Luebke 7 01/14/19OpenGL: Specifying Normals●Calling glColor() sets the color for all vertices following, until the next call to glColor()●Calling glNormal() sets the normal vector for the following vertices, till next glNormal() ●So flat-shaded lighting requires:glNormal3f(Nx, Ny, Nz);glVertex3fv(v0);glVertex3fv(v1);glVertex3fv(v2);■While smooth shading requires:glNormal3f(N0x, N0y, N0z); glVertex3fv(v0);glNormal3f(N1x, N1y, N1z); glVertex3fv(v1);glNormal3f(N2x, N2y, N2z); glVertex3fv(v2);■(Of course, lighting requires additional setup…)David Luebke 8 01/14/19More OpenGL●Other things you’ll need to know:■To clear the screen:glClearColor(r, g, b, a);glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);●Nate Robins has an excellent set of OpenGL tutorials that help illustrate many of these concepts & functions:http://www.cs.utah.edu/~narobins/opengl.html Also http://xmission.com/~nate/tutors.html ●Next: the math and concepts underlying the transformation callsDavid Luebke 9 01/14/19Translations●For convenience we usually describe objects in relation to their own coordinate system●We can translate or move points to a new position by adding offsets to their coordinates:■Note that this translates all points uniformlyzyxtttzyxzyx'''David Luebke 10 01/14/19Scaling●Scaling a coordinate means multiplying each of its components by a scalar●Uniform scaling means this scalar is the same for all components: 2David Luebke 11 01/14/19Scaling●Non-uniform scaling: different scalars per component:●How can we represent this in matrix form?X 2,Y 0.5David Luebke 12 01/14/19Scaling●Scaling operation:●Or, in matrix form:czbyaxzyx'''zyxcbazyx000000'''scaling matrixDavid Luebke 13 01/14/192-D Rotation(x, y)(x’, y’)x’ = x cos() - y sin()y’ = x sin() + y cos()(Draw it)David Luebke 14 01/14/192-D Rotation●This is easy to capture in matrix form:●3-D is more complicated■Need to specify an axis of rotation■Simple cases: rotation about X, Y, Z axes yxyxcossinsincos''David Luebke 15 01/14/193-D Rotation●What does the 3-D rotation matrix look like for a rotation about the Z-axis?■Build it coordinate-by-coordinatezyxzyx1000)cos()sin(0)sin()cos('''David Luebke 16 01/14/193-D Rotation●What does the 3-D rotation matrix look like for a rotation about the Y-axis?■Build it coordinate-by-coordinatezyxzyx)cos(0)sin(010)sin(0)cos('''David Luebke 17 01/14/193-D Rotation●What does the 3-D rotation matrix look like for a rotation about the X-axis?■Build it coordinate-by-coordinatezyxzyx)cos()sin(0)sin()cos(0001'''David Luebke 18 01/14/193-D Rotation●General rotations in 3-D require rotating about an arbitrary axis of rotation●Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra●Standard approach: express general rotation as composition of canonical rotations ■Rotations about X, Y, ZDavid Luebke 19 01/14/19Composing Canonical Rotations●Goal: rotate about arbitrary vector A by ■Idea: we know how to
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