48-175 Descriptive Geometry Spatial Relations on Lines 1 A line - PowerPoint PPT Presentation

48-175 
 Descriptive Geometry Spatial Relations on Lines 1

A line is parallel to a plane if it has no common point with the plane. To test whether a given line and plane are parallel : simply, construct an edge view of the plane and project the line into the same view; if the line appears in point view or parallel to the edge view , then it cannot meet the plane in a point, and is therefore parallel to the plane This fact can be used to construct a plane parallel to a given line or a line parallel to a given plane. Lines parallel to a plane

3 3 3 3 1 1 1 1 B B B B M M A A A A O O O O Two possible lines parallel to Two possible lines parallel to Two possible lines parallel to B B B B B plane ABC through point O plane ABC through point O plane ABC through point O N N M M A A A A A C C C C O O O O O Line through O parallel to the Line through O parallel to the Line through O parallel to the Line through O parallel to the C C C C C edge view of plane ABC edge view of plane ABC edge view of plane ABC edge view of plane ABC N N 1 1 1 1 1 2 2 2 2 2 C C C C C N How do we locate How do we locate A A A A A points M and N? points M and N? O O O O O line parallel through a point B B B B B parallel to a plane M M M M M

B How do we determine if two planes are parallel ? E by constructing an auxiliary view that shows A one plane in edge view; if the other plane is also seen in edge view then the two planes are parallel C D F 1 2 C F A D B parallel planes E

E B B 1 3 1 3 Constructing an auxiliary view that shows one plane A in edge view; if the other plane is also seen in edge B B D view then the two planes are parallel E E A A C C F parallel edge views indicate parallel planes C C D D F F 1 1 2 2 C C F F A A D D B B parallel planes E E

A line is perpendicular to a N Q plane if every line in the B L M plane that passes through P the point of intersection of O the given line and the plane Lines LM, NO, PQ all lie in the plane makes a right angle with the given line Line AB is perpendicular to the plane A line perpendicular to plane (normal)

p N M p is a plane Line AB is a normal to it Q P Lines LM, NO and PQ lie in the plane A,B L O 1 2 L,P B M,Q B L,O M,N 90º 90º A A 2 3 perpendicular line to plane (normal)

3 3 1 1 direction of the direction of the normal in view #1 normal in view #1 normal is perpendicular normal is perpendicular to the edge view of plane to the edge view of plane 90º 90º TL TL 1 1 1 2 2 2 direction of the normal to a plane direction of the direction of the direction of the normal in view #2 normal in view #2 normal in view #2

direction of the direction of the direction of the normal in view #1 normal in view #1 normal in view #1 90º 90º TL TL Two-view method to find 1 1 1 direction ( bearing ) 2 2 2 TL 90º direction of the direction of the direction of the direction of the normal to a plane normal in view #2 normal in view #2 normal in view #2

C C C C P P P P B B B B A A A A 1 1 1 1 2 2 2 2 B B B B A A A A P P P C C C C quiz: perpendicular to the plane at point P

O Shortest line (OP) from point O to plane ABC A F C P E Lines AD and EF D lie in the plane ABC B Observer's line of sight – plane ABC is seen as an edge and true length of OP appears shortest distance from a point and a plane

C C C True length of the shortest True length of the shortest True length of the shortest line from M to the plane line from M to the plane line from M to the plane 3 3 3 1 1 1 M M M P P P A A h h M M M Edge view Edge view Edge view C C C of plane ABC of plane ABC of plane ABC B B B P P B B B P lies on the perpendicular from M P lies on the perpendicular from M A A A A to the true length line in view #1 to the true length line in view #1 1 1 1 1 2 2 2 2 B B B h h M M M A A A A P is located by using the P is located by using the P P transfer distance from view #3 transfer distance from view #3 or by tracing a line on the or by tracing a line on the plane through P plane through P C C C C

how do we determine if a plane is perpendicular to a given plane ? this requires finding edge views of the plane and seeing if they are perpendicular to each other – which we will consider it later when we consider lines of intersection perpendicular planes

revisiting an old problem – shortest distance to a line

As line AB is in true length, the constructed As line AB is in true length, the constructed perpendicular from X to AB produces point Y perpendicular from X to AB produces point Y X X X 4 3 4 3 X X B B B True length of the True length of the Y TL TL TL A A A shortest distance shortest distance X X X X B B B B AB,Y AB,Y 3 3 3 Point view of line AB Point view of line AB 1 1 1 Y Project back from view #3 to get Y A A A A 1 1 1 1 2 2 2 2 Y Project back from view #1 to get Y A A A A constructing shortest distance to a line (line method) X X X X

3 3 4 4 X X X X X Edge view of Edge view of Edge view of ABX ABX ABX True shape of True shape of ABX ABX 3 3 3 1 1 1 A A A Project back from view #4 to get Y X Y is the shortest X Y is the shortest distance from X to AB distance from X to AB B B B X X X X B B B B A A A A 1 1 1 1 ABX defines a plane ABX defines a plane ABX defines a plane ABX defines a plane 2 2 2 2 B B B B Project back from view #1 to get Y A A A A X X X X constructing shortest distance to a line (plane method)

B Line XY is the shortest distance between skew lines AB and CD X as it is perpendicular to both lines 90° D 90° A Y C shortest distance between skew lines

B D A C 1 2 D B C A shortest distance between skew lines

4 3 AB is in true length in view #3 B X A AB, X 90° C C 90° D Y Y D B D True length of shortest line XY is seen in view #4 X 3 Common perpendicular XY 1 between skew lines AB and A Y CD in view #1 C 1 2 D Y B Common perpendicular XY between C skew lines AB and CD in view #2 shortest distance between X skew lines ( line method ) A

B B D D Y W A A X C C 1 1 2 2 HL D B B W X C C A A shortest distance between skew lines ( plane method )

B D 4 R,S 3 3 1 C B XY is parallel to AB in view #1 D, Y and passes through W A B D R Shortest distance RS between Y skew lines AB and CD R S A W A C,X S Plane CXYD seen in X C edge view in view #3 1 2 HL Y D CXYD is a plane XY is parallel to AB in view #2 S B and meets CD at W W X DY is parallel to folding line 1|2 C shortest distance between skew lines ( plane method ) R A

Horizontal projection plane X Y parallel X Y Shortest horizontal distance between the two skew lines shortest horizontal distance between skew lines

A D B C 1 2 B C shortest horizontal distance D between skew lines A

A D LM is parallel to CD in view #1 A M XY is parallel to the edge view of B the horizontal plane XY is also the true length of the TL BL is in D L shortest horizontal line M true length C 1 CD is parallel to the edge B,L 2 view of plane ABLM in C view #3 HL View #3 is an elevation L B LM is parallel to M C CD in view #2 shortest horizontal distance between skew lines A

B A X,Y A D 1 C 3 X D 4 3 LM is parallel to CD in view #1 A M Y X XY is parallel to the edge view of B the horizontal plane XY is also the true length of the TL BL is in D L shortest horizontal line M true length C Y 1 CD is parallel to the edge B,L 2 view of plane ABLM in C view #3 HL View #3 is an elevation L B LM is parallel to M C CD in view #2 X Y shortest horizontal distance D between skew lines A

B D X,Y A 1 A C 3 X D LM is parallel to CD 4 in view #1 A 3 M X 15° N 4 Y B O 4 TL BL is in D L M true length C XY is also the true length of the Y shortest upward 15° grade line 1 B,L 2 C HL View #3 is an elevation L B LM is parallel to M C CD in view #2 Y D X shortest grade distance between skew lines A

B D A 1 C 3 A 90° 4 3 X D LM is parallel to CD in view #1 A 20% grade M Y X B TL BL is in D L M true length C Y 1 XY is also the true length of the B,L 2 shortest downward 20%grade line C HL View #3 is an elevation L B LM is parallel to M C CD in view #2 X Y D which grade distance? A

Observers line of sight in which line AB is above line CD D A B C visibility

l l l 1 1 t t B B B midpoint midpoint TL TL X X B B B B A A A l l l l Project back from view #1 to get X X A A A A t t t t f f f f B B B B X X l l l l Project back from top view to get X A A A A quiz: find a point on a line equidistant to two points

quiz: locating a line between two skew lines through a point