Viewing and Camera Rotations CSE 5542 Prof. Roger Crawfis

Introduction to 3D viewing 3D is just like taking a photograph!

Viewing Transformation Position and orient your camera

Projection Transformation Control the “lens” of the camera Project the object from 3D world to 2D screen

Viewing Transformation (2) Important camera parameters to specify Camera (eye) position (Ex,Ey,Ez) in world coordinate system Center of interest (coi) (cx, cy, cz) Orientation (which way is up?) View-up vector (Up_x, Up_y, Up_z) world (cx, cy, cz) (ex, ey, ez) view up vector (Up_x, Up_y, Up_z)

Viewing Transformation (3) Form a camera (eye) coordinate frame Transform objects from world to eye space world u v n x y z Eye coordinate frame coi

Viewing Transformation (4) Eye space? Transform to eye space can simplify many downstream operations (such as projection) in the pipeline (1,0,0) (0,1,0) u v (0,0,1) n y world (0,0,0) coi x z

Viewing Transformation (5) In OpenGL: - gluLookAt (Ex, Ey, Ez, cx, cy, cz, Up_x, Up_y, Up_z) - The view up vector is usually (0,1,0) - Remember to set the OpenGL matrix mode to GL_MODELVIEW first

Viewing Transformation (6) void display() { glClear(GL_COLOR_BUFFER_BIT); glMatrixMode(GL_MODELVIEW); glLoadIdentity(); gluLookAt(0,0,1,0,0,0,0,1,0); display_all(); // your display routine } If this is constant, why do it every frame?

Suppose we have three orthogonal vectors… v1,v2,v3 Let’s build a matrix like this: This will rotate: v1 onto the x axis, v2 onto the y axis, v3 onto the z axis

My Version of gluLookAt() void mygluLookAt(Point3 eye, Point3 center, Point3 up) { Point3 cameraZ = Normalize( eye – center ); // v3 Point3 cameraX = Normalize( Cross(up, cameraZ) ); // v1 Point3 cameraY = Cross( cameraZ, cameraX ); // v2 GLdouble m[16]; // Fill the matrix in row by row m[0] = cameraX.X; m[4] = cameraX.Y; m[8] = cameraX.Z; m[12] = 0.0; m[1] = cameraY.X; m[5] = cameraY.Y; m[9] = cameraY.Z; m[13] = 0.0; m[2] = cameraZ.X; m[6] = cameraZ.Y; m[10] = cameraZ.Z; m[14] = 0.0; m[3] = m[7] = m[11] = 0.0; m[15] = 1.0; glMultMatrixd( m ); glTranslated( -eye.X, -eye.Y, -eye.Z ); } Order of transformations!

Messing with the camera What if I want to PAN the camera? We need to define “PAN” There are two possibilities Crab Pan

Panning Suppose we pan around the camera Y axis This is NOT “up” in world space. We need to change the lookAt point.

Operations on Points in Space LookAt is a point in space Need these transformations: Translate by –eye Rotate frame onto axis (using some M) Rotate around Y by pan angle Inverse rotate M Translate by eye PP=T(eye) MT RY(q) M T(-eye)

Easier #1 Just replace the matrix using gluLookAt. Problems?

Easier #2 The first 2 operations are already what is done to set the camera up: M T(-eye) We just need to add a rotate after this is done. Implies we want to pre-multiply by a rotation matrix.

Easier #2 Steps: Read out the current matrix. Set the matrix to the identity matrix. Rotate about the y-axis the amount we want to pan. Post-multiply by the matrix read out in step #1. With GLSL Shaders, the OpenGL matrix system does not buy you much, so just use your own matrix classes.

Camera Controls Tilt Roll Dolly Boom Zoom (same as dolly?) Eliot Lash, 2007 (from Wikipedia.org – Camera dolly) Tilt Roll Dolly Boom Zoom (same as dolly?) General camera (or entity) movement and the user interface / control.

Camera Controls Stationary 2 degrees of freedom + zoom QuicktimeVR

Interactive Applications How do we add interactive control? Many different paradigms Examiner => Object in hand Fly-thru => In a virtual vehicle pod Walk-thru => Constrained to stay on ground. Move-to / re-center => Pick a location to fly to. Collision detection? Can we pass thru objects like ghosts?

Interactive Applications What do we use to control the motion? Mouse One-button, two-button, three-button What button does what? Only when mouse is clicked down, released up, or continuously as the mouse moves? Keyboard Arrow keys?

Input Devices Interactive user control devices Mouse 3D pointer - Polhemus, Microscribe, … Spaceball Hand-held wand Data Glove Gesture Custom

A Virtual Trackball A rather standard and easy-to-use interface. Examiner type of interaction. Consider a hemi-sphere over the image-plane. Each point in the image is projected onto the hemi-sphere.

A Virtual Trackball Points inside the projection of the hemi-sphere are mapped up to the surface. Determine distance from point (mouse position) to the image-plane center. Scale such that points on the silhouette of the sphere have unit length. Add the z-coordinate to normalize the vector.

A Virtual Trackball Do this for all points. Keep track of the last trackball (mouse) location and the current location. This is the direction we want the scene to move in. Take the direction perpendicular to this and use it as the axis of rotation. Use the distance between the two points to determine the rotation angle (or amount).

A Virtual Trackball Rotation axis: Where, v1 and v2 are the mouse points mapped to the sphere. v2 | sin q| = v1

Virtual Trackball How to calculate p1 and p2? Assuming the mouse position is (x,y), then the sphere point P also has x and y coordinates equal to x and y Assume the radius of the hemi-sphere is 1. So the z coordinate of P is Note: normalize viewport y extend to -1 to 1 If a point is outside the circle, project it to the nearest point on the circle (set z to 0 and renormalize (x,y)) z y (x,y,0) x

Virtual Trackball Visualization of the algorithm

Example Example from Ed Angel’s OpenGL Primer In this example, the virtual trackball is used to rotate a color cube The code for the colorcube function is omitted I will not cover the following code, but I am sure you will find it useful Note you can use the trackball to move the camera or you can apply the rotations to a transform node in your scene graph to move an object. The rotations are negated from each other though.

Initialization #define bool int /* if system does not support bool type */ #define false 0 #define true 1 #define M_PI 3.14159 /* if not in math.h */ int winWidth, winHeight; float angle = 0.0, axis[3], trans[3]; bool trackingMouse = false; bool redrawContinue = false; bool trackballMove = false; float lastPos[3] = {0.0, 0.0, 0.0}; int curx, cury; int startX, startY;

The Projection Step void trackball_ptov(int x, int y, int width, int height, float v[3]) { float d, a; /* project x,y onto a hemisphere centered within width, height , note z is up here*/ v[0] = (2.0*x - width) / width; v[1] = (height - 2.0F*y) / height; d = sqrt(v[0]*v[0] + v[1]*v[1]); v[2] = cos((M_PI/2.0) * ((d < 1.0) ? d : 1.0)); a = 1.0 / sqrt(v[0]*v[0] + v[1]*v[1] + v[2]*v[2]); v[0] *= a; v[1] *= a; v[2] *= a; }

glutMotionFunc (1) Void mouseMotion(int x, int y) { float curPos[3], dx, dy, dz; /* compute position on hemisphere */ trackball_ptov(x, y, winWidth, winHeight, curPos); if(trackingMouse) /* compute the change in position on the hemisphere */ dx = curPos[0] - lastPos[0]; dy = curPos[1] - lastPos[1]; dz = curPos[2] - lastPos[2];

glutMotionFunc (2) if (dx || dy || dz) { /* compute theta and cross product */ angle = 90.0 * sqrt(dx*dx + dy*dy + dz*dz); axis[0] = lastPos[1]*curPos[2] – lastPos[2]*curPos[1]; axis[1] = lastPos[2]*curPos[0] – lastPos[0]*curPos[2]; axis[2] = lastPos[0]*curPos[1] – lastPos[1]*curPos[0]; /* update position */ lastPos[0] = curPos[0]; lastPos[1] = curPos[1]; lastPos[2] = curPos[2]; } glutPostRedisplay();

Idle and Display Callbacks void spinCube() { if (redrawContinue) glutPostRedisplay(); } void display() glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT); if (trackballMove) glRotatef(angle, axis[0], axis[1], axis[2]); colorcube(); glutSwapBuffers();

Mouse Callback void mouseButton(int button, int state, int x, int y) { if(button==GLUT_RIGHT_BUTTON) exit(0); /* holding down left button allows user to rotate cube */ if(button==GLUT_LEFT_BUTTON) switch(state) case GLUT_DOWN: y=winHeight-y; startMotion( x,y); break; case GLUT_UP: stopMotion( x,y); }

Start Function void startMotion(int x, int y) { trackingMouse = true; redrawContinue = false; startX = x; startY = y; curx = x; cury = y; trackball_ptov(x, y, winWidth, winHeight, lastPos); trackballMove=true; }

Stop Function void stopMotion(int x, int y) { trackingMouse = false; /* check if position has changed */ if (startX != x || startY != y) redrawContinue = true; else angle = 0.0; redrawContinue = false; trackballMove = false; }

public void KeyTrackballArrows( object sender, Forms public void KeyTrackballArrows( object sender, Forms.KeyEventArgs e ) { // This provides a simple mapping of the keyboard to rotations of the // object (or the world if used in a camera). // The SHIFT key can be used for gross changes, the CTRL key can be // used for fine-control. Vector3 rotationAxis = XAxis; float sign = 1.0f; if( e.KeyCode == Keys.Up ) { rotationAxis = XAxis; sign = -1.0f; } else if( e.KeyCode == Keys.Down ) { rotationAxis = XAxis; else if( e.KeyCode == Keys.Left ) { rotationAxis = YAxis; sign = -1.0f; else if( e.KeyCode == Keys.Right ) { rotationAxis = Yaxis; else { return; float rotationAngle = RotationIncrementStandard; if( e.Modifiers == Keys.Shift ) rotationAngle = RotationIncrementLarge; else if( e.Modifiers == Keys.Control ) rotationAngle = RotationIncrementSmall; if( rotate != null ) { rotate( sign*rotationAngle, rotationAxis ); }