CS 351: Perspective Viewing

Transcription

1 CS 351: Perspective Viewing Instructor: Joel Castellanos Web: 2/16/2017 Perspective Projection 2 1

2 Frustum In computer graphics, the viewing frustum is the three-dimensional region which is visible on the screen which is formed b a clipped pramid. 3 Properties of Perspective Projections Propert 1: The perspective projection of an object becomes smaller as the object gets farther awa form the center of projection. What are some non-obvious, and interesting effects of this? 4 2

3 Properties of Perspective Projections Propert 2: As an object is rotated, its projected width becomes smaller. This is known as foreshortening. 5 Properties of Perspective Projections Propert 3: Perspective projections preserve straight lines. 6 3

4 Properties of Perspective Projections Propert 4: Sets of parallel lines that are parallel to the view plane remain parallel when projected onto the view plane. Propert 5: Sets of parallel lines that are not parallel to the view plane converge to a vanishing point on the view plane. 7 Equation of a Checkerboard //Checkerboard in x-z plane with = 0. public static Color getcheckerboardgroundcolor( double x, double, double z) { if ( Math.abs((Math.floor(x))) % 2 == Math.abs((Math.floor(z))) % 2) return Color.BLACK; return Color.WHITE; } 8 4

5 Axis-Aligned Perspective Projection Straight down Y-Axis Ee: (0, 10, 0) One Point on View Plane: (x vp, 5, z vp ) (0, 10, 0) ra ee d( ViewPlanePt ee) xhit 0 xvp d 5 10 zhit 0 zvp d (5 10 ) 0 d ee ( d ( ee ee vp vp ) ee ) x z hit hit x z ee ee d ( x d ( z vp vp x z ee ee ) ) 9 A Practical Viewing Sstem The virtual pinhole camera implements perspective viewing with the following features: An arbitrar ee point. An arbitrar view direction (The view plane is defined as being perpendicular to the view direction and centered on the ra from the ee point). An arbitrar orientation about the view direction. An arbitrar distance between the ee point and the view plane. 10 5

6 Phsical Pinhole Camera Clear inverted image with small pinhole Fuzz out-of-focus image with larger hole 11 Lens Aperture Aperture Shutter Aperture Shutter Wh color reflections? 12 6

7 Large Glass is Expensive Canon, Prime 50mm f/1.8 USM Lens: $ Canon, Prime 50 mm f/1.4 USM Lens: $ Canon, Prime 50 mm f/1.2 USM Lens: $1, Depth of Field 14 7

8 Circle of Confusion 15 Real lenses do not focus all ras perfectl. Thus, at best focus, a point is imaged as a spot rather than a point. The smallest such spot that a lens can produce is often referred to as the circle of least confusion. Perspective Views of Boxes How man vanishing points are there in each image? In each image, does the view direction point up or down or is it horizontal? How can ou tell when it's horizontal? 16 a b c 8

9 Quiz In the equation: a b c / b c a, b and c are vectors. This means the have both magnitude and direction. What can be said about the magnitude and direction of a? 17 Virtual Pinhole-Camera Viewing Sstem 18 User Input: The ee point, e. The look-at point, l. The up vector, up. The view-plane distance d. 9

10 Primar-Ra Calculation w v v w ( e l) u up w v w u e l up w 19 The (x v, v ) coordinates of a sample point p on the pixel in row r and column c are: The primar-ra direction d is: x v v s( c hres / 2 px) s( r vres / 2 p ) d x u v dw v v 3D Translation A translation is a geometric transformation that moves ever point of a figure or a space b the same amount in a given direction. A translation transformation does not change the object's shape or size. dx x dx x d d dz z dz z 20 10

11 3D Scaling Uniform scaling is a linear transformation that enlarges (increases) or shrinks (diminishes) objects b a scale factor that is the same in all directions. The result of uniform scaling is similar (same shape different size) to the original. r 0 0x rx 0 r 0 r 0 0 r z rz Uniform scaling rx 0 0x rxx 0 r 0 r 0 0 rz z rz z Non-uniform scaling 21 3D Rotation about the x-axis x Rx( ) 0 cos sin 0 sin cos z 1x00z x 0 x (cos ) (sin ) z (cos ) (sin ) z 0 x (sin ) (cos ) z (sin ) (cos ) z 22 11

12 3D Rotation about the -axis and z-axis cos 0 sin x R( ) sin 0 cos z cos sin 0x Rz( ) sin cos z 23 12