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

48-175 
 Descriptive Geometry

Spatial Relations on Lines

1

Lines parallel to a plane 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.

2 1

M B C A O O C A B

1 3

Line through O parallel to the edge view of plane ABC 2 1

M O B A C B C A O O C A B

1 3

Two possible lines parallel to plane ABC through point O Line through O parallel to the edge view of plane ABC 2 1

M O B A C B C A O O C A B

How do we locate points M and N?

1 3

Two possible lines parallel to plane ABC through point O Line through O parallel to the edge view of plane ABC 2 1

M N M O B A C B C A O O C A B M N

How do we locate points M and N?

1 3

Two possible lines parallel to plane ABC through point O Line through O parallel to the edge view of plane ABC 2 1

M N N M O B A C B C A O O C A B M N

line parallel through a point parallel to a plane

parallel planes How do we determine if two planes are parallel ?

2 1

D E E F F B C A D C A B

by constructing an auxiliary view that shows

• ne plane in edge view; if the other plane is

also seen in edge view then the two planes are parallel

parallel planes

Constructing an auxiliary view that shows one plane in edge view; if the other plane is also seen in edge view then the two planes are parallel

1 3 2 1

D E E F F C B B C A D C A B

parallel edge views indicate parallel planes 1 3 2 1

D D E E F F F C A B B C A D C A B E

line perpendicular to plane (normal) A line is perpendicular to a plane if every line in the plane that passes through the point of intersection of the given line and the plane makes a right angle with the given line

Line AB is perpendicular to the plane Lines LM, NO, PQ all lie in the plane

B A P Q L M N O

perpendicular line to plane (normal)

p 90º 90º

3 2 2 1 p is a plane Line AB is a normal to it Lines LM, NO and PQ lie in the plane

A B M,N L,O B M,Q L,P P Q A,B L M N O A

direction of the normal to a plane

2 1

direction of the normal in view #2

TL

normal is perpendicular to the edge view of plane

1 3 2 1

direction of the normal in view #1

90º

direction of the normal in view #2

TL

normal is perpendicular to the edge view of plane

1 3 2 1

direction of the normal in view #1

90º

direction of the normal in view #2

direction of the normal to a plane

direction of the normal in view #2 direction of the normal in view #1

2 1 TL

direction of the normal in view #2

90º

direction of the normal in view #1

1 2 TL TL 2 1

direction of the normal in view #2 direction of the normal in view #1

90º 90º

Two-view method to find direction (bearing)

quiz: perpendicular to the plane at point P

1 2

B A C

P

A C B

1 2

P

B A C

P

A C B

1 2

P

B A C

P

A C B

1 2

P

B A C

P

A C B

shortest distance from a point and a plane

Observer's line of sight – plane ABC is seen as an edge and true length of OP appears Shortest line (OP) from point O to plane ABC Lines AD and EF lie in the plane ABC

P A C B O D E F

2 1

A A C

Edge view

• f plane ABC

True length of the shortest line from M to the plane 2 1 1 3

P M C B M C A B M A C B h h

Edge view

• f plane ABC

True length of the shortest line from M to the plane P is located by using the transfer distance from view #3

• r by tracing a line on the

plane through P 2 1 1 3 P lies on the perpendicular from M to the true length line in view #1

P P P M C A B M C A B M A C B h h

Edge view

• f plane ABC

True length of the shortest line from M to the plane P is located by using the transfer distance from view #3

• r by tracing a line on the

plane through P 2 1 1 3 P lies on the perpendicular from M to the true length line in view #1

P P P M C A B M C A B M A C B

perpendicular planes 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 how do we determine if a plane is perpendicular to a given plane ?

revisiting an old problem – shortest distance to a line

constructing shortest distance to a line (line method)

1 2

A X A B X

TL

3 1 1 2

X A B

A X A B X

TL

4 3 As line AB is in true length, the constructed perpendicular from X to AB produces point Y Point view of line AB True length of the shortest distance 3 1 1 2

X AB,Y X A B

A X A B X

TL

Project back from view #1 to get Y Project back from view #3 to get Y 4 3 As line AB is in true length, the constructed perpendicular from X to AB produces point Y Point view of line AB True length of the shortest distance 3 1 1 2

Y Y Y X AB,Y X A B

A X A B X

constructing shortest distance to a line (plane method)

ABX defines a plane 1 2

A B X A B X

ABX defines a plane Edge view of ABX 3 1 1 2

A X B A B X A B X

ABX defines a plane X Y is the shortest distance from X to AB 4 3 Edge view of ABX True shape of ABX 3 1 1 2

A X X B A B X A B X

ABX defines a plane X Y is the shortest distance from X to AB Project back from view #4 to get Y 4 3 Project back from view #1 to get Y Edge view of ABX True shape of ABX 3 1 1 2

A X X B A B X A B X

shortest distance between skew lines

A B D C

Line XY is the shortest distance between skew lines AB and CD as it is perpendicular to both lines 90° 90°

X Y

shortest distance between skew lines

2 1

A B C D B A D C

shortest distance between skew lines (line method)

90° 90°

Common perpendicular XY between skew lines AB and CD in view #1 AB is in true length in view #3 Common perpendicular XY between skew lines AB and CD in view #2 True length of shortest line XY is seen in view #4 4 3 3 1 2 1

X Y X X Y Y Y D C AB, X C D A B A B C D B A D C

shortest distance between skew lines (plane method)

2 1 A B C D B A D C

HL

2 1 Y X W W X A B C B A D C

shortest distance between skew lines (plane method)

HL

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

shortest horizontal distance between skew lines

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

X Y Y X

shortest horizontal distance between skew lines

1 2

B A D C A B D C

shortest horizontal distance between skew lines

HL TL

View #3 is an elevation XY is parallel to the edge view of the horizontal plane XY is also the true length of the shortest horizontal line CD is parallel to the edge view of plane ABLM in view #3 BL is in true length LM is parallel to CD in view #1 LM is parallel to CD in view #2

2 1

M A D C B,L L M L M C A B A B D C

shortest horizontal distance between skew lines

HL TL

View #3 is an elevation XY is parallel to the edge view of the horizontal plane XY is also the true length of the shortest horizontal line

3

CD is parallel to the edge view of plane ABLM in view #3

4 1 3

BL is in true length LM is parallel to CD in view #1 LM is parallel to CD in view #2

2 1

M X Y Y X Y X X,Y B C A D A D C B,L L M L M C D A B A B D C

shortest grade distance between skew lines

HL TL

15° View #3 is an elevation XY is also the true length of the shortest upward 15° grade line

3 4 1 3

BL is in true length LM is parallel to CD in view #1 LM is parallel to CD in view #2

2 1

Y X Y X Y X X,Y B C D A O4 M A D C B,L L M L M C D A B A B D C N4

which grade distance?

HL TL

90° 20% grade View #3 is an elevation XY is also the true length of the shortest downward 20%grade line

3 4 1 3

BL is in true length LM is parallel to CD in view #1 LM is parallel to CD in view #2

2 1

X Y Y X Y X C D B A M A D C B,L L M L M C D A B A B D C

visibility

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

quiz: find a point on a line equidistant to two points

l l

f t

A B B A l l l

f t

B A A B B A l l

TL

l Project back from top view to get X Project back from view #1 to get X

1 t f t

X X X

midpoint

B A A B B A l l

TL

l

1 t f t

X X

midpoint

B A A B B A

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

quiz: construct a line at a certain grade

Lines AB and CD specify centerlines of two existing sewers as shown in the figure. The sewer pipes are to be connected by a branch pipe having a downward grade of 2:7 from the higher to the lower pipe. Given that point C is 20' North of point A, the problem is to determine the true length and bearing of the branch pipe and show this pipe in all views. Line AC (in plan) measures 20'.

20'

f t

Y X Y

C D B D C A A B

20'

1 t

X

D C B,Y A,X

Y

C D B A TL = 30'-8"

bearing = S 51.5° E 2 1 2:7 grade

D C B A D C B,Y A,X

20'

bearing = S 51.5° E f t

X Y Y

C D B D C A A B

quiz: construct a line at a certain grade

quiz: shortest distance from X to nearest face

5'-0" 4'-0" 3'-6" 1'-6" 1'-0" 45° X X

a f t

X X

a edge view of face a t 1 f t X1 X X

Y is not on face p ∴ need to find point on face a nearest Y in the same plane

Y is the foot of the perpendicular from X to the plane of face p p edge view of face p t 1 f t Y Y X X X

p Z is projected back from view #1 Z is projected back from view #2 true length of AB Z is perpendicular from XY to AB in view #2 2 1 t 1 f t Z Z Z X,Y A Y Y B A X X X A B

= 2'-1" true shortest distance between X and Z is given in view #3 3 t p 2 1 t 1 f t Z X Z Z Z X,Y A Y Y B A X X X A B

Problem

Three equal legs of a surveyor’s tripod are located in their relationship to the plumb line. Leg A bears N30°W and has a slope of 30° Leg B is 3’-3” due east of the plumb line and at the same elevation as the plumb line Leg C bears S45°W and has a slope of 45° The plumb bob touches the bench mark at a vertical distance of 4’ below the top

• f the line

Determine TL of legs A, B and C ? What is the angle B makes with plumb line? Show legs in front and top views.

Step 1

Step 2

Problem Two sewer lines AB and CB converge at manhole B A is 35’ north 10’ east of B and 30’ above B C is 20’ north 60’ west of B and 15’ above B A new line DB is located in the plane of ABC at a point 30’ due west of A Using only two views locate D What is the TL of each sewer line? What is the angle of the plane ABC

Step 1

Step 2

Step 3

Problem AB is parallel to plane. 
 Complete the view.

Step

Problem Planes ABC and DEF are parallel. Complete the view

Step

What is the shortest length between non- intersecting diagonals of adjacent faces of a cube?

Problem Given two partial pipelines – construct the shortest level pipeline to connect them. These pipelines may be extended

Step 1

Step 2

Step 3

Problem involving perpendicular lines Construct the pyramid with base A(2, 2, 5), B(3.5, 1.5, 6), C(4.75, 2, 6) D(3.25, 2.5, 5) height 2.5 All measurements in inches Complete top and front views with proper visibility ie., visible lines are solid and not visible lines shown dashed.

What is the clearance between a pipe and the ball?