Viewing and Camera Rotations

CSE 781Prof. Roger Crawfis

Introduction to 3D viewing

• 3D is just like taking a photograph!

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Viewing Transformation

• Position and orient your camera

3

Projection Transformation

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

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Viewing Transformation (2)

• Important camera parameters to specify

5

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)

• Transformation? – Form a camera (eye) coordinate frame

– Transform objects from world to eye space 6

world

uv 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

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world

uv n

x

y

z

(0,0,0) coi

(1,0,0)(0,1,0)(0,0,1)

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

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Viewing Transformation (6)

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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

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1000000

),,(,3,3,3

,2,2,2

,1,1,1

321zyx

zyx

zyx

vvvvvvvvv

vvvR

My Version of gluLookAt()

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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

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Panning

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

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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)

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Easier #1

• Just replace the matrix using gluLookAt.– Problems?

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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.

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Easier #2

• Steps:1. Read out the current matrix.2. Set the matrix to the identity matrix.3. Rotate about the y-axis the amount we want to

pan.4. Post-multiply by the matrix read out in step #1.

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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?)• General camera (or entity) movement and the

user interface / control.

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Eliot Lash, 2007 (from Wikipedia.org – Camera dolly)

Camera Controls

• Stationary• 2 degrees of freedom +

zoom• QuicktimeVR

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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 projectedonto 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 tonormalize 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 themouse points mappedto the sphere.

v1

v2

1 2u v v

| sin q| = ||||

||

21 ppn

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))

22 yx1

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 Stepvoid 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 Callbacksvoid 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 Callbackvoid 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); break;

} }

Start Functionvoid 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 Functionvoid 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.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 );    }}