画法几何：第四讲平面的投影Projection of Plane.ppt

第四讲 平面的投影 Projection of Plane 平面的表示法 Representations of planes 各种位置的平面 Position of Planes 平面上的点和直线Points and lines in planesZXY平面图形投影总是为平面图形或直线 The projection of a planar surface always is a line or a planar surface.1、平面的表示法 Representations of planesabcabc不在同一直线上的三个点 Three points not on a straight lineabcabc直线及线外一点 A line and a point not on the lineabcabcdd两平行直线 Two parallel linesabcabc两相交直线Two intersecting linesabcabc平面图形 Planar shape投 影 特 性 平面平行投影面-投影体现实形 Plane Projection plane 平面垂直投影面-投影积聚成直线 Plane Projection plane 平面倾斜(Inclined)投影面-投影亲似原平面 Plane inclined to Projection plane2、各种位置的平面 Positions of Planes平行Parallel垂直perpendicular倾斜inclined存真性True shape亲似性Similarity积聚性Collecting(1) 平面对一个投影面的投影特性Projection characteristics of planes to a single projection plane(2) 平面在三投影面体系中的投影特性平面对于三投影面的位置可分为三类：Positions relative to three principal projection planes投影面垂直面Planes perpendicular to projection plane投影面平行面Planes parallel to projection plane一般位置平面General position plane特殊位置平面Special position planes垂直于某一投影面，倾斜于另两个投影面Perpendicular to a projection plane and inclined to the other two平行于某一投影面，垂直于另两个投影面Parallel to a projection plane and perpendicular to the other two与三个投影面都倾斜Inclined to three projection planes正垂面V-perpendicular侧垂面W-perpendicular铅垂面H-perpendicular 正平面 Frontal plane 侧平面 Profile plane 水平面 Horizontal planeOXYZHVaAaWabB bbV 正垂面 V-perp. plane(3) 投影面垂直面Planes perpendicular to projection planeCcccH 铅垂面 H-perp. planeW 侧垂面 W-perp. planeplumb line ：a cord from which a metal (lead) weight is suspended pointing directly to the earths center of gravity; used to determine the vertical from a given point铅 垂 面 H-perp. planeLeadPerpendicularPlaneabcacbcba亲似性Affinity亲似性Affinity积聚性collecting铅垂面H-perpendicular plane投影特性： 在它垂直的投影面上的投影积聚成直线。

该直线与投影轴的夹角反映空间平面与另外两投影面夹角的大小 It appears as a line in the perpendicular projection plane. The angles between that line and the projection axes represented the angles between the plane and the other two projection planes. 另外两个投影面上的投影有亲似性The other two projections are affinal shapes.例1. 已知多边形的VH投影，求其W投影Complete the W projection of polygon with the given V & H projections.aba (b)cdd(c)efghg(h)f(e)abcdefghabc abcabc(4) 投影面平行面Planes parallel to projection plane积聚性collecting积聚性collecting存真性True shape水平面Horizontal plane投影特性：(1)一个投影存真 One projection is true-show type.(2)另外两投影积聚-面平行于投影轴的线 The other two projections are line-collecting type. (Plane- line axis)(3)积聚性与存真性同时存在 Both collecting and true-show typeOXYZHVaAaCWabBbcccb(5)一般位置平面General position plane/Oblique plane3D图形与2D投影有亲似性2D projections is affinal to 3D shape OXYZHVaaWabbcccb一般位置平面投影 Projections of General position planeY上行平面 Ascending plane下行平面 Descending plane一般位置平面投影 Projections of General position plane平面的位置判断 POSITION OF PLANES一般位置平面General-position plane投影面垂直面Planes perpendicular to a projection planeH perpendicular plane投影面垂直面Planes perpendicular to a projection plane投影面平行面Planes parallel to a projection plane3、平面上的点和直线Points and lines in planes判断直线在平面内的方法How to determine if a line in a plane 定 理 一Theorem 1 若一直线过平面上的两点，则此直线必在该平面内。

A line is on a plane if it goes through two points belonging to the plane定 理 二Theorem 2 若一直线过平面上的一点，且平行于该平面上的另一直线，则此直线在该平面内 A line will be on a plane if it goes through one point in the plane and parallel to a line lying in the plane(1) 平面上取任意直线 Locate arbitrary line in plane线在面上的条件Line located in the plane if: (1) 过面上的两点 Through two points in the plane (2) 过面上一点, 且平 行于面上另一直线 Through one point of the plane and parallel to one line in the planeABALabcbcaabcbcadmnnmd例2：已知平面由直线AB、AC所确定，试在平面内任作一条直线。

Construct a line arbitrarily in plane defined by lines AB and AC.解法一Solution1解法二Solution2根据定理二By Theorem 2根据定理一By Theorem 1例3：在平面ABC内作一条水平线，使其到H面的距 离为10mm Construct a horizontal line 10mm away from H projection plane in plane ABCnmnm10cabcab 唯一解！The exclusive solution!点在面上的几何条件: 必在面内的一条直线上Point lies in a plane if it lies on a line in the plane.(2) 平面上取点 Locating a point in a plane 先找出过此点而又在平面内的一条直线作为辅助线，然后再在该直线上确定点的位置 A point can be located in a plane by first constructing a auxiliary line on the plane through the point. 面上取点的方法：面上取点必先取线 Locating a point in plane must be by first locating a line.步骤Steps:(1) 在P面内过A点作直线L Draw line L on P through point A.例4: A是平面P上一点, 已知A点的V投影, 求H投影AP, find the H projection of point A.分析: 若A点在平面P上, 必在过A点的直线L上(L在P上) If AP then A L (L P )pp2112llLPaaA(2) 作直线L的H投影Draw H projection of line L.(3) 根据从属性作出直线L上的A点的H投影Get the H projection of point A.例5：已知K点在平面ABC上，求K点的水平投影。

K plane ABC , find the H projection of point K.baccakbkabcabkcdkd利用平面的积聚性求解By converging plane通过在面内作辅助线求解By auxiliary linebckadadbcadadbckbc例6：已知AC为正平线，补全平行四边形ABCD的水平投影ACV, Complete the H projection of parallelogram ABCD解法一Solution 1解法二Solution 2(3)过点或直线作平面Construct plane through point or line过空间中任意一点A作平面Construct plane through point AL过空间中任意一直线L作平面Construct plane through line L结论: 要有约束条件才能得到唯一解Conclusion: Exclusive solution must with extra constraintsA 过直线L作平面Construct plane through line Lllaammllaamm 过点A作平面Construct pl。