1、平面體與回轉(zhuǎn)體相貫 Penetration between plane solid and revolved solid,回轉(zhuǎn)體與回轉(zhuǎn)體相貫 Penetration between two revolved solids,復(fù)合相貫 Composite penetration,一、概述,1.相貫的形式 Various penetrations,兩立體相交叫作相貫，其表面產(chǎn)生的交線叫做相貫線。 Penetration is intersection of two solids. The intersection lines are on the surfaces of the solids,第十五

2、講 立體與立體相交（相貫） Intersection of solids ( Penetration),2.相貫線的主要性質(zhì) Intersection lines of solids,其作圖實(shí)質(zhì)是找出相貫的兩立體表面的若干共有點(diǎn)的投影。To construct the intersection lines is to find the common points of the surfaces., 共有性 Common to both solids, 表面性 On surfaces of two solids,相貫線位于兩立體的表面上。,相貫線是兩立體表面的共有線。, 封閉性 Close sp

3、atial curves,相貫線一般是封閉的空間折線（通常由直線和曲線組成）或空間曲線。,二、平面立體與回轉(zhuǎn)體相交（作業(yè)P42-11） Intersection of plane solid and revolved solid,1.相貫線的性質(zhì) The intersection lines,相貫線是由若干段平面曲線（或直線）所組成的空間折線，每一段是平面體的棱面與回轉(zhuǎn)體表面的交線。 The intersection lines are spatial polylines composed of several plane lines which are the intersections o

4、f planes and revolved surfaces.,2.作圖方法, 分析各棱面與回轉(zhuǎn)體表面的相對(duì)位置，從而確定交線的形狀。, 求出各棱面與回轉(zhuǎn)體表面的截交線。, 連接各段交線，并判斷可見性。,求交線的實(shí)質(zhì)是求各棱面與回轉(zhuǎn)面的截交線。,1. 相貫線的性質(zhì) The intersection,相貫線一般為光滑封閉的空間曲線，它是兩回轉(zhuǎn)體表面的共有線。The intersection usually is smooth and close spatial curve which is the common of the surfaces of two solids.,三 、 回轉(zhuǎn)體與回轉(zhuǎn)體

5、相貫 Intersection of two revolved solids,2.作圖方法Method, 利用投影的積聚性直接找點(diǎn)。Finding common points by concentrative projection., 用輔助平面法。By auxiliary plane method, 先找特殊點(diǎn)。Find special points fisst., 作圖過程Steps, 補(bǔ)充中間點(diǎn)。Add some in-between points.,確定交線的彎曲趨勢(shì) Define the trend of the curving intersection,確定交線的范圍 Define

6、 the limits of the intersection,例 2 ：圓柱與圓柱相貫(正交相貫)，求其相貫線。 Finding the intersection of two cylinders,相貫線的已知投影 find the given views of intersection,利用積聚性，采用表面取點(diǎn)法。By Concentrative projections, 找特殊點(diǎn) Special points, 補(bǔ)充中間點(diǎn)Other points, 光滑連接Connecting smoothly,1”,2”,3”,3,3,a”,a,a,例 2：圓柱與圓柱相貫，求其相貫線。Finding

7、the intersection of two cylinders,作題步驟: 找到相貫線的已知投影 Find the given projections 面上找點(diǎn)（先特殊點(diǎn)，后中間點(diǎn)）Find intersection points 順序、光滑連接各點(diǎn) Connect points smoothly 分析輪廓線 The contour lines 判斷可見性 Visibility,3“,5,a,b,2,3,4,5,6,1,7,3“,5,4,2,3,4,5,6,a,b,例 3：圓柱與圓柱相貫（偏貫），求其相貫線。Finding the intersection of two cylinders

8、,1,2,3,4,a,5,6,7,b,2“,6,a,b“,a“,b“,2“,6,4,1“,7,所在兩立體表面都可見交線才可見 Visible only if both of surfaces are visible,例 3：圓柱與圓柱相貫，求其相貫線。Finding the intersection of two cylinders,相貫線和回轉(zhuǎn)體的輪廓線是相切關(guān)系（邊界定理） Intersection curve should be tangent to the contours,例 4：圓柱與圓錐相貫，求其相貫線的投影。 Complete the top view and front vi

9、ew of intersection of cone and cylinder., 空間及投影分析Analysis in space and projections：,相貫線為一光滑的封閉的空間曲線。它的側(cè)面投影有積聚性，正面投影、水平投影沒有積聚性，應(yīng)分別求出。Intersection is a smooth ,close 3-D curve which concentrates on side view while top view and front view are not concentrated., 解題方法：輔助平面法 Auxiliary plane method,例 4：圓柱與

10、圓錐相貫，求其相貫線的投影。 Complete the top view and front view of intersection of cone and cylinder.,假想用水平面P截切立體，P面與圓柱體的截交線為兩條直線，與圓錐面的交線為圓，圓與兩直線的交點(diǎn)即為交線上的點(diǎn)。 Cut the solids imaginarily with horizontal plane P which intersects with cylinder at two lines and intersects with cone at a circle. The intersection point

11、s of two lines and circle belong to the intersection curve.,解題步驟Step：, 求特殊點(diǎn)Special Points, 用輔助平面法求中間點(diǎn) Other points by auxiliary plane method, 光滑連接各點(diǎn) Connect smoothly,例 4：圓柱與圓錐相貫，求其相貫線的投影。 Complete the top view and front view of intersection of cone and cylinder.,例 4：圓柱與圓錐相貫，求其相貫線的投影。 Complete the to

12、p view and front view of intersection of cone and cylinder.,四 兩回轉(zhuǎn)體相貫線的變化形式 All types of intersections of two revolved solids,1、當(dāng)圓柱直徑變化時(shí)，相貫線的變化趨勢(shì) The intersection change with the radius of cylinder,交線為四次空間曲線交線向大圓柱軸線彎曲,交線為兩條平面曲線（橢圓） （蒙若定理）,2、當(dāng)其中一個(gè)圓柱不貫穿時(shí)： One cylinder stops while intersects the other o

13、ne,3、蒙若定理Monge Theorem 兩個(gè)二次曲面若與另一個(gè)二次曲面外切（或內(nèi)切），則相交于二次曲線。Two conicoids intersects at conic plane curve if both of them are tangent to a third conicoid,柱柱相貫,一個(gè) m次曲面與一個(gè)n次曲面相交，交線（相貫線）為mn次曲線。One m surface intersects with an n surface , and their intersection will be mn curve.,Gaspard. Monge 法國人(1764-1818),畫法幾何