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View in 2D & 3D
Teacher: A.prof. Chengying Gao(高成英)
E-mail: [email protected]
School of Data and Computer Science
Computer Graphics
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Outline
• 2D Viewing
• 3D Viewing
• Classic view
• Computer view
• Positioning the camera
• Projection
2Computer Graphics
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2D Viewing
3Computer Graphics
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2D Viewing
4Computer Graphics
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Windowing Concepts
5Computer Graphics
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2D Viewing Transformation
6Computer Graphics
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Deriving Viewport Transformation
7Computer Graphics
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Deriving Viewport Transformation
8Computer Graphics
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The Aspect Ratio (纵横比)
9Computer Graphics
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Maintaining the Aspect
10Computer Graphics
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OpenGL Commands
11Computer Graphics
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OpenGL Commands
12Computer Graphics
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Outline
• 2D Viewing
• 3D Viewing
• Classic view
• Computer view
• Positioning the camera
• Projection
13Computer Graphics
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3D Viewing
14Computer Graphics
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The Pinhole Camera
15Computer Graphics
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Transformation and Camera Analogy
16Computer Graphics
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Outline
• 2D Viewing
• 3D Viewing
• Classic view
• Computer view
• Positioning the camera
• Projection
17Computer Graphics
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Classic View
18Computer Graphics
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Plane Geometry Projection
• That is projected onto the plane of the standard projection
• Projection line is a straight line
• Gathering in the center of projection
• Parallel to each other
• This projection preserve collinearity
• Some application such as mapping (地图映射)need to be
non planar projection
19Computer Graphics
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Taxonomy of Projection
20Computer Graphics
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Orthogonal projection (正交投影)
• Projection line perpendicular (垂直) to the plane of projection
21Computer Graphics
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Multi-view orthographic projection (多角度正交投影)
• The projection plane parallel to the reference plane(DOP is perpendicular to the view plane)
• Usually projection from the front, top and side
22Computer Graphics
Front
BackTop
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Multi-view of CAD parts
23Computer Graphics
在CAD和建筑行业中,通常
显示出来三个视点图以及等
角投影图。
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Advantages & Disadvantages
• Keep the distance and angle
• Remain the shapes
• Use for measurement (building, manual)
• Can't see the global real object shape, because many surface
not visible in view
• Sometimes adding isometric drawing (等角图)
24Computer Graphics
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Axonometric projection (轴测投影)
25Computer Graphics
等轴测 正二测 正三测
三个主面对称 两个主面对称 三个面比率不同
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Mechanical Drawing
26Computer Graphics
不等角图等角图
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Oblique Projections (斜平行投影)
27Computer Graphics
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Perspective projection
28Computer Graphics
PRP: Projection Reference Point = Eye position
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Early Perspective
29Computer Graphics
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Vanishing Points
• On the object of all parallel lines (not parallel to the projection plane) projected to a point
• Hand draw simple perspective projection on the need to use the vanishing point
30Computer Graphics
Vanishing Point
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Vanishing Points
31Computer Graphics
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Perspective Projection
32Computer Graphics
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Perspective Viewing
33Computer Graphics
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34Computer Graphics
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35Computer Graphics
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36Computer Graphics
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Outline
• 2D Viewing
• 3D Viewing
• Classic view
• Computer view
• Positioning the camera
• Projection
37Computer Graphics
The fundamental difference between the
classic view and computer view:
• All the classical views are based on a particular
relationship among the objects (对象), the
viewers (观察者), and the projectors (投影线).
• In computer graphics, we stress the
independence of the object specifications (对象
定义) and camera parameters (摄像机参数).
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Computer view
• The view has three functions, are implemented in pipeline
system
• Positioning the camera
• Setup the model-view matrix
• Set the lens
• Projection matrix
• Clipping
• view frustum
38Computer Graphics
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Camera in OpenGL
• In OpenGL, the initial world frame and camera frame are the
same
• A camera located at the origin, and point to the negative
direction of Z axis
• OpenGL also specifies the view frustum default, it is a center at
the origin of the side length of 2 cube
39Computer Graphics
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Default projection
• Default is the projection of orthogonal projection
40Computer Graphics
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Outline
• 2D Viewing
• 3D Viewing
• Classic view
• Computer view
• Positioning the camera
• Projection
42Computer Graphics
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Moving the camera frame
• If you want to see objects with positive Z coordinate more, we
can
• Moves The camera along the positive Z axis
• Moves the object along the negative Z axis
• They are equivalent, is determined by the model-view matrix
• Need a translation: glTranslated(0.0, 0.0, d);
• Here, d > 0
43Computer Graphics
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Moving the camera frame
44Computer Graphics
Default frame
After translated d, d>0
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Moving the camera frame
• Can use a series of translation and rotation to the camera
position to any position
• For example, in order to get the side view
• Rotate the camera: R
• Move the camera from the origin: T
• C = TR
45Computer Graphics
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Viewing Specification
46Computer Graphics
正方向
观察参考点
观察方向
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View Reference Coordinate System
47Computer Graphics
通过观察平面法向量和观
察正向向量,确定摄像机。
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View Reference Coordinate System
49Computer Graphics
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Simplest Camera Position
50Computer Graphics
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World to Viewing coordinate Transformation
• The world could be transformed so that the view reference
coordinate system coincides with the world coordinate system
• Such a transformation is called world to viewing coordinate
transformation
• The transformation matrix is also called view orientation matrix
51Computer Graphics
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Deriving View Orientation Matrix
52Computer Graphics
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A More Intuitive Approach Offered by GLU
53Computer Graphics
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OpenGL Viewing Transformation
54Computer Graphics
它封装了世界坐标系到观察坐标系的转换。调用之后,我们就把坐标
系变换的矩阵放入了矩阵栈,后续对物体的位置描述,会通过此矩阵
栈进行转换到我们的观察坐标系了。
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gluLookAt Illustration
55Computer Graphics
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Look-At Positioning
56Computer Graphics
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Where does it point to?
57Computer Graphics
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Constructing a Coordinates
58Computer Graphics
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Rotation
59Computer Graphics
view reference frame:
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Translation
60Computer Graphics
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Composing the Result
61Computer Graphics
viewing coordinate transformation is:
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The Viewing Transformation
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Looking At a cube
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gluLookAt() and other transformations
• The user can define the model-view matrix to achieve the
same function
• But from the concept of the gluLookAt () as the camera
position, while the other follow-up transformation as
object position
• gluLookAt in the OpenGL () function is the only specialized
for positioning the camera function
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Outline
• 2D Viewing
• 3D Viewing
• Classic view
• Computer view
• Positioning the camera
• Projection
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Orthogonal Projection
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Orthogonal Projection Matrix: Homogeneous coordinates
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Perspective Projection
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Transformation Matrix for Perspective Projection
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Perspective Projection
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Basic Perspective Projection
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Homogeneous Coordinates
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Perspective Divide
• 如果𝑤 ≠ 1,那么必须从齐次坐标中除以𝑤而得到所表示的
点
• 这就是透视除法,结果为
上述方程称为透视方程。
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Perspective Projection
• Perspective Divide 是非线性的,导致非均匀缩短。
• 离投影中心(COP)远的对象投影后,尺寸缩短得比离COP近的对
象大。
• 透视变换是保直线的,但不是仿射变换。
• 透视变换是不可逆的,因为沿一条投影直线上的所有点投
影后的结果相同。
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Perspective Projection
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View Window
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View Volume & Clipping
• For perspective projection the view volume is a semi-infinite
pyramid with apex (顶点) at prp and edges passing through the
corners of the view window
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• For efficiency, view volume is
made finite by specifying the
front and back clipping plane
specified as distance from the
view plane
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View Volume & Clipping
• For parallel projection the view volume is an infinite parallelepiped
(平行六面体) with sides parallel to the direction of projection
• Clipping is done in 3D by clipping the world against the front clip
plane, back clip plane and the four side planes
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• View volume is made finite by specifying
the front and back clipping plane specified
as distance from the view plane
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The Complete View Specification
• Specification in world coordinates
• position of viewing (vrp),
direction of viewing(-n),
• up direction for viewing
(upVector)
• Specification in view coordinates
• view window : center (cx, cy),
width and height,
• prp : distance from the view
plane,
• front clipping plane : distance
from view plane
• back clipping plane : distance
from view plane
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Orthogonal view in OpenGL
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Perspective in OpenGL
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Perspective in OpenGL
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Field of application
• Application of glFrustum sometimes difficult to get the desired results
• GluPerspective (fovy, aspect, near, far) can provide a better results
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Aspect = w / h
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Normalization
85Computer Graphics
• Normalization allows for a single pipeline for both perspective
and orthogonal viewing.
• It simplifies clipping.
• Projection to the image plane is simple (discard z).
• z is retained for z-buffering (visible surface determination)
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Transformation Pipeline
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1. Vertices of the Object to draw are in Object space (as modelledin your 3D Modeller)
2. … get transformed into World space by multiplying it with the Model Matrix
3. Vertices are now in World space(used to position the all the objects in your scene)
4. … get transformed into Camera space by multiplying it with the View Matrix
5. Vertices are now in View Space– think of it as if you were looking at the scene throught “the camera”
6. … get transformed into Screen space by multiplying it with the Projection Matrix
7. Vertex is now in Screen Space –This is actually what you see on your Display.
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Projective Rendering Pipeline
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