yv xv zv

yv xv zv

yv xv zv

Ex.กำหนดให้วัตถุมีจุดยอดเป็น จงหาด้าน Front และ side view yv 7 12 11 10 6 9 7 12 4 8 3 5 xv 1 2 zv

yv 7 12 10 11 6 7 4 9 3 8 12 1 2 5 xv Only one face of an object zv

These two planes are called the near-far clipping planes, or the front-back clipping planes. The near and far planes allow us to exclude objects that are in front of or behind the part of the scene that we want to display.

The orthogonal-projection view volume

Axonometric Orthogonal Projection We can also form orthogonal projections that display more than one face of an object. Such views are called axonometric orthogonal projections. The most commonly used axonometric projection is the isometric projection, which is generated by aligning the projection plane (or the object) so that the plane intersects each coordinate axis in which the object is defined, called the principal axes.

Isometric Projection y y x x z z

Isometric Projection 1) Rotation about the y-axis 2) Rotation about the x-axis 3) An orthogonal projection to the z=0 plane

2) 1)

3)Pz

Ex.กำหนดให้วัตถุมีจุดยอดเป็น จงหา isometric projection 7 12 yv 11 10 6 9 7 12 4 8 3 5 xv 1 2 zv

yv 6 7 11 10 8 5 9 3 12 4 2 xv 1 zv

7-7 Oblique Parallel Projections In general, a parallel-projection view of a scene is obtained by transferring object descriptions to the view plane along projection paths that can be in any selected direction relative to the view-plane normal vector. When the projection path is not perpendicular to the view plane, this mapping is called an oblique parallel projection.

Cavalier projection of the unit cube onto the z=0 plane [Foley et al.] Cabinet Projection

P’ L P R S

1) Shearing of the object in space in a direction parallel to the plane of projection 2) Orthogonal projection onto the plane of projection

1) Shearing 2) Projection Pz

Ex วัตถุ P(3,4,0), Q(1,0,4), R(2,0,5) และ S(4,0,3) โปรเจ็คชันแบบ Oblique (Cavalier) ลงบนระนาบ Z=0 โดยมุมแอลฟา = 45 องศา, L=1 yv P Q xv S R zv

7-8 Perspective Projections We can approximate this geometric-optics effect by projecting objects to the view plane along converging paths to a position called the projection reference point (or center of projection).

Perspective projection y z-axis vanishing point x z

One-Point Perspective Projection [Foley et al.] fig 6.3 One-point perspective projections of a cube onto a plane cutting the z axis showing vanishing point of lines perpendicular to projection plane

2-Point Perspective Projections 1) Rotate object with an angle about y-axis 2) Translate object by [0 m n] 3) Perform 1-point perspective projection onto the xy-plane (z=0)

Rotate object with an angle about y-axis

Translate object by [0 m n]

1-Point Perspective projection onto Z=0

Ex. ลูกบาศก์ถูกกระทำแบบ 2-point perspective projection โดยการหมุนรอบแกน y ไปเป็นมุม 30 องศา แล้วเลื่อนไปด้วยระยะ (0,3,-3) และ center of projection คือ (0,0,2) y Zcp x 4 3 5 8 2 1 6 7 z

y x z

3-Point Perspective Projections 1) Rotate object with an angle about y-axis 2) Rotate object with an angle about x-axis 3) Translate object by [0 m n] 4) Perform 1-point perspective projection onto the xy-plane (z=0)