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1、Viewing In 3DChapter 5n 如何在二维平面上显示三维物体如何在二维平面上显示三维物体n 表示问题表示问题n 遮挡关系的反映遮挡关系的反映n 真实感问题真实感问题投影三维形体的表示消隐光照与着色The basic question of 3D graphics三维图形学的基本问题3D viewing processClip againstviewvolumeProject ontoprojectionplaneTransform into viewport in 2D device coordinates for display3D world-coordinate outp
2、ut primitivesClipped world coordinates2D device coordinates primitivesFig.Conceptual model of the 3D viewing process.Conception Projections transform points in a coordinate system of dimension n into points in a coordinate system of dimension less than n.Projection(投影）Define:The projection of a 3D o
3、bject is defined by straight projection rays(projectors投影线投影线)emanating from a center of projection（投影投影中心）中心）,passing through each point of the object,and intersecting a projection plane（投影面）（投影面）to form the projection.ProjectionProjections Parameter:n projection center(投影中心投影中心)n projection plane(
4、投影面）投影面）平面几何投影：平面几何投影：投影面是平面，投影线为直线。投影面是平面，投影线为直线。The class of projectionsn Perspective Projection(透视投影透视投影)The distance between the center of projection and the projection plane is finite.(投影中心到投影面之间的距离是有限的)nParallel Projection(平行投影平行投影)The distance between the center of projection and the projecti
5、on plane is infinite.(投影中心到投影面之间的距离是无限的)The class of projectionsS S(a)(a)透视投影透视投影S S(b)(b)平行投影（正投影）平行投影（正投影）S S(c)(c)平行投影平行投影（斜投影）斜投影）投影可分为两大类：投影可分为两大类：n透视投影透视投影:投影中心到投影面之间的距离是有限的投影中心到投影面之间的距离是有限的n平行投影平行投影:投影中心到投影面之间的距离是无限的投影中心到投影面之间的距离是无限的The class of projectionsABABCenter of projectionprojectorsPr
6、ojection plane(a)Fig.(a)perspective projection (b)parallel projection.ABABCenter of projectionprojectorsProjection plane(b)The subclasses of Planar Geometric Projections.平面几何投影平面几何投影平行投影平行投影透视投影透视投影正投影正投影斜投影斜投影三视图三视图正轴测正轴测斜等测斜等测斜二测斜二测正等测正等测正二测正二测正三测正三测正视图正视图侧视图侧视图俯视图俯视图一点透视一点透视二点透视二点透视三点透视三点透视正投影图正投
7、影图投影方向投影方向垂直于垂直于投影面的投影面的平行投影。分为平行投影。分为三视图三视图和和正轴正轴侧图侧图。当投影面与某一坐标轴垂当投影面与某一坐标轴垂直时，得到的投影为直时，得到的投影为三视图三视图；否则得到的投影为否则得到的投影为正轴侧图正轴侧图。(b)(b)正轴测图正轴测图xyO O投影面投影面投影方向投影方向z投影方向投影方向投影面投影面(a)(a)三视图三视图zxyThree Orthographic ProjectionsProjectors for top view（俯视图）Projection plan (front view)Projectors for side view
8、（侧视图）Projectors for front view（主视图）Projection plan (top view)Projection plan (side view)Three Orthographic Projections(三视图）VUXYYZ主视图侧视图俯视图YZX正平行投影（三视图）VUXYYZ主视图侧视图俯视图txtztxtytytz(a,b)Front Elevation10000000100001zxtbtaTVUXYYZ主视图侧视图俯视图txtztxtytytzTop ElevationVUXYYZ主视图侧视图俯视图txtztxtytytz10000000010001
9、yxtbtaTSide ElevationVUXYYZ主视图侧视图俯视图txtztxtytytz10000000100010zytbtaT正轴侧图正轴侧图(投影面不与坐标轴垂直时的正投影投影面不与坐标轴垂直时的正投影)n正轴侧有正轴侧有等轴侧等轴侧、正二侧正二侧和和正三侧正三侧三种。三种。n等轴侧等轴侧：投影面与三个坐标轴：投影面与三个坐标轴之间的夹角都相等；之间的夹角都相等；n正二侧正二侧：投影面与两个坐标轴：投影面与两个坐标轴之间的夹角相等时；之间的夹角相等时；n正三侧正三侧：投影面与三个坐标轴：投影面与三个坐标轴之间的夹角都不相等时。之间的夹角都不相等时。(b)(b)正轴测图正轴测图xy
10、O O投影面投影面投影方向投影方向z斜平行投影斜平行投影n定义定义：投影方向与投影平面不：投影方向与投影平面不垂直时的平行投影。垂直时的平行投影。n斜投影图斜投影图(即斜轴测图即斜轴测图):投影投影方向不垂直于投影面所得到的方向不垂直于投影面所得到的平面投影。平面投影。n常用的斜轴测图有常用的斜轴测图有斜等测图斜等测图和和斜二测图斜二测图。S S(c)(c)平行投影平行投影（斜投影）斜投影）投影平投影平面法向面法向投影方向投影方向投影平面投影平面(a)(a)斜等测斜等测a ap pO Opp(b)(b)斜二测斜二测投影方向投影方向投影平面投影平面a ap pO Opp投影平投影平面法向面法向对
11、于对于斜等测图斜等测图有：有：=45,ctg=1斜二测图斜二测图则有：则有：=arctg(2),ctg=1/2The subclasses of Planar Geometric Projections.平面几何投影平面几何投影平行投影平行投影透视投影透视投影正投影正投影斜投影斜投影三视图三视图正轴测正轴测斜等测斜等测斜二测斜二测正等测正等测正二测正二测正三测正三测正视图正视图侧视图侧视图俯视图俯视图一点透视一点透视二点透视二点透视三点透视三点透视透视投影透视投影n定义定义：n投影中心与投影平面距离为有限远（此时投影投影中心与投影平面距离为有限远（此时投影线汇聚于投影中心）线汇聚于投影中心）n
12、特点特点：n真实感强真实感强n近大远小近大远小n不与投影面平行的平行线经投影后汇聚于一点不与投影面平行的平行线经投影后汇聚于一点Perspective Projectionn Vanishing Point（灭点）灭点）The perspective projections of any set of parallel lines that are not parallel to the projection plane converge to a vanishing point.（任何一束不平行于投影平面的平行线的透视任何一束不平行于投影平面的平行线的透视投影将汇聚于一点，称为灭点）投影将汇
13、聚于一点，称为灭点）nAxis Vanishing Point（主灭点）（主灭点）If the set of lines is parallel to one of the principal axes,the vanishing point is called an axis vanishing point.（平行于平行于坐标轴的平行线对应的灭点）坐标轴的平行线对应的灭点）The style of Perspective Projection Perspective projections are categorized by their number of principal vanish
14、ing points and therefore by the number of axes the projection plane cuts nOne-point perspective projection（一点透视）一点透视）Projection-plane cuts only one of three axes.（有一个主灭点，即投影面与一个坐标轴正交，与另外两个坐标轴平行）nTwo-point perspective projection（两点透视）两点透视）Projection-plane cuts only two of three axes at the same time.
15、（有两个主灭点，即投影面与两个坐标轴相交，与另一个坐标轴平行）nThree-point perspective projection（三点透视）三点透视）Projection-plane cuts all of three axes at the same time.（有三个主灭点，即投影面与三个坐标轴都相交）The style of Perspective ProjectionOne-point perspective projectionFig.Construction of one-point perspective projection of cube onto plane cutti
16、ng the z axis.Projection-plane normal is parallel to z axis.xzyCenter of projectionProjection plane normalProjection planeTwo-point and Three-point Perspective ProjectionFig.Two-point and three-point perspective projection of a cube.(a)Two-point perspective,the projection plane cuts the x and z axes.(b)Three-point perspective,the projection plane cuts three axes at the same time.(a)xzy(b)tzzzztyyyytxxxxccsccsccs)()()(The mathematics of simple Perspective ProjectionParameter Equation of Line PcP:
