Descriptive Geometry Eng. Areej Afeefy Ref. Descriptive Geometry Metric PARE/LOVING/HILL Fifth edition

Descriptive Geometry Descriptive Geometry: is the science of graphic representation and solution and space problems By Arch. Areej Afeefy

projections Two common types of projections: 1) perspective projections (used by architects or artists) 2) orthographic projections (perpendicular to the object)

Perspective Projection Screen Human eye

Orthographic Projection Screen

Orthographic Projection Horizontal Front Profil

Horizontal Profil Front

Principal Views

Draw the profile

Steps to obtain a view Establish the line of sight. Introduce the folding line Transfer distances to the new view Determine visibility and complete the view

Primary Auxiliary Views Step 1: Establish the line of sight. Primary Auxiliary Views Step 2: Introduce the folding line Step 3: transfer distances a,e d,h y b,f c,g H 1 Step 4: determine visibility and complete view 1 D2 D1 D y h d g k1 e D2 c y f a f,e g,h D1 b b,a c,d

All views projected from top view has the same height dimension

Primary Auxiliary Views

View 1 is an auxiliary view projected from the front View

All the views projected from front view have the same depth dimension

Edge View of a plane

Chapter 3: LINES

Frontal Line

Frontal Line

the true angel between a line and any projection plane appears in any view shows the line in true length and the projection plane in Edge View.

Level (Horizontal) Line

Level Line

Profile Line

Profile Line

True Length of an Oblique Line

True Length of an Oblique Line

Bearing , Slope, and Grade aH aH S55oE 55 b b Bearing: a term used to describe the direction of a line on the earth’s surface

Azimuth Bearing N aH aH 125o N125o b b

problem A 160-m segment AB of a power line has a bearing of N 60o and a downward slope of 20o from the given point A. Complete the front and top views. ah aF

D1 b 1 20o H 160 m N b N 60o a1 D1 ah H F D1 D1 aF b

Grade Grade: another way to describe the inclination of a line from the horizontal Plane

Grade

Chapter 4 Planes

Points and lines in Planes

Locating a Point in a Plane Problem: Given the front and side views of a plane MON and the front view of a point A in the plane. Determine the side view n n m m aF x oF oP

Solution n n m Y m Y aF x aPx X X oF oP

Lines in Planes b g c aH e b aF c e Complete the front view

Lines in Planes b x g c aH e b aF x g c e

Principal Lines in Planes

Frontal Line All frontal lines in the same plane are parallel unless the plane it self is frontal

Horizontal or Level Lines

Horizontal or Level Lines All horizontal lines in the same plane are parallel unless the plane it self is horizontal

Profile Line

Profile Line All profile lines in the same plane are parallel unless the plane it self is profile

Locus The Locus: is the path of a point, line or curve moving is some specified manner. Or it is the assemblage of all possible positions of a moving point, line or curve The locus of a point moving in a plane with a specified distance from another point is circle.

Locus Problem: in the given plane ABC locate a point K that lies 6 mm above horizontal line AB and 5 mm in front of frontal line AC. Scale: full size

Solution h c aH K 5 mm f f h b c f h h K 6 mm aF f b

Pictorial Intersection B H D N E C K M Two principles to solve the problem: 1) Lines in a single plane must either be parallel or intersect. 2) If two planes are parallel, any lines on the planes in question are parallel.

Pictorial Intersection

Pictorial Intersection

Successive Auxiliary Views Chapter 5 Successive Auxiliary Views

Construction of successive Auxiliary Views Step 1: Establish the line of sight. Step 2: Introduce the necessary folding lines. Step 3: transfer distance to the new view. Step 4: Complete view.

Point View of a Line A line will appear in point view if the line of sight is parallel to the line in space.. In the drawing sheet, the line of sight should be parallel to the true length of the line.

Point View of a Line 2 b a2 ah b H a1,b F b T.L. Point View (P.V) aF 1

Problem I Find the true clearance between the point O and the line AB. 1 a2,b Clearance b o o ah T.L. 2 H o F aF ah b o

Edge View of a Plane A plane will appear in edge view in any view for which the line of sight is parallel to the plane. In the drawing sheet, a plane will appear in edge view in any view for which the line of sight is parallel to a true length line in the plane.

Edge View of a Plane c c h ah E.V. b a H F b c h b T.L. 1 aF

Normal Views of a Plane A normal view or TRUE SIZE and shape of a plane is obtained in any view for which the line of sight is perpendicular to the plane. In the drawing sheet the line of sight appear perpendicular to the Edge View of the plane.

Edge View of a Plane c 2 c h E.V. ah b a H b F c h b T.L. 1 aF Normal View T.S. b a H F b c h b T.L. 1 aF

Uses of Auxiliary and additional Views Position of line of sight In space On the drawing sheet 1) True length of line (TL) Perpendicular to line Perpendicular to any view of the line or directed to a point view of the line 2) Point view of line Parallel to line Parallel to the true length of the line 3) Edge view of plane (EV) Parallel to plane Parallel to true length of line in plane OR directed toward a true size view of plane 4) Normal or true size view of plane (TS) Perpendicular to plane Perpendicular to edge view of plane

problem Find the front and top views of a 2.5m radius curve joining two intersecting lines BA & BC.

b a f b c c a c c b b TL f a a

b a 2 1 3 5 4 3 f 4 b c 2 5 c 1 a c c 1 1 2 2 b 3 b 3 TL f 4 4 5 5 a a

Chapter 6 Piercing Points

Piercing point The intersection of a line with a plane is called Piercing Point. If the line is not in or parallel to a plane, it must intersect the plane.

Piercing point - Auxiliary View Method 1. e b1,c c p e TL bH a p g g a a e p g c bF

Piercing point- Two View Method A piercing point could be found using the given views as follows: (see the following Fig.) Any convenient cutting plane containing line EG is introduced, it appears EV in a principal view. The line of intersection between the two planes is determined. Since line EG and line 1 - 2 both lies in the cutting plane they intersect, locating point P. Since line 1 – 2 also lies in Plane ABC, point P is the required Piercing Point.

Piercing point- Two View Method A Vertical cutting plane N 1 E P G C 2 B

Piercing point- Two View Method a e 1 p c Vertical cutting plane N 2 bH g a 1 e p g c bF 2

Intersection of Planes Chapter 7 Intersection of Planes

Intersection of Planes Any two planes either parallel or must intersect. Even the intersection beyond the limits of planes. The intersection of planes result a line common to both of them.

Intersection of Planes Auxiliary view Method bH e x k z a b1 f y k e g J x c a y j a c k g e c x z y f J g bF

Intersection of Planes Auxiliary view Method bH e k z a bH y k e g J x c a y j a c k c g e z y J g bF

Intersection of Planes Two View - Piercing point Method b b d d a a x x g g y y eF eP cF cP

Intersection of Planes Two View - Piercing point Method b E.V. b d d L1 L1 2 a 2 a x x E.V. 1 1 g g 4 y y 4 3 3 eF eP cF cP

Intersection of Planes Two View - Piercing point Method b b d d a a g g eF eP cF cP

Intersection of Planes Cutting Plane Method Line of intersection m c H1 P1 2 3 1 4 H2 6 7 P2 5 b 8 n a o

Intersection of Planes Cutting Plane Method cH m P1 2 1 3 4 6 P2 7 8 b nH LI o 5 a cF m 1 2 EV of HI P1 4 3 5 6 8 EV of H2 a P2 7 o b nH LI

Pictorial Intersection of Planes 3 a d s c k b e m 2 o

Pictorial Intersection Of Planes 3 a v c k b m 2

Chapter 8 Angle between Planes

Angle between Planes B θ m E.V. of m θ n E.V. of n A P.V. of line of intersection AB Line of sight

Dihedral Angle Line of Intersection given eH B e1 A TL LI eF g B LI e2g E.V. of A E.V. of B θ g

Dihedral Angle Line of Intersection is NOT given 4 o x 3 2 m kH y bH c 1 a EV.1 o n 3 4 x bF EV.2 2 y m 1 kF c

Dihedral Angle Line of Intersection is NOT given b2 a n o X,y x m c θ kH n y bH m c a x o b1 a n TL o n x y k1 bF m y m kF c c

Dihedral Angle Line of Intersection is NOT given Alternative solution: You can find the Edge View for both planes without resorting to find the line of intersection. See next slide

Dihedral Angle Line of Intersection is NOT given Both Planes will Appear EV. a b2 n o a TS m kH o bH c c k2 TL n 3 a o n m 2 bF TL m kF n c m o k1 EV a b1,c 1

Angle between Oblique Plane and Principal Plane EV of frontal plane b aH f c H F 1 F c a1 b θf c f TL b aF Angle between plane and frontal plane

Angle between Oblique Plane and Principal Plane aH c H F P 1 c c TL b b aF f f b aP EV of Profile plane θP c a1 Angle between plane and Profile plane

Angle between Oblique Plane and Principal Plane Angle between a plane and a horizontal plane can be measured in the similar fashion. The angle between sloping plane and a horizontal plane is called DIP ANGLE.

Angle between Oblique Plane and Principal Plane 1 aH b b θH aH TL c f c H F c f b aF Angle between plane and horizontal plane

Chapter 9 Parallelism

Parallel Lines Oblique Lines that appears parallel in two or more principal views are parallel in space.

Parallel Lines d b c aH H P F aF aP b b c c d d

Parallel Lines c b d aH H F F P aF b aP b c d d c

Principal Line Two horizontal, two frontal, or two profile lines that appears to be parallel in two principal views may or may not be parallel in space. non intersecting, non parallel lines are called SKEW LINES.

Parallel Lines aH X a1b e c X b 1 X c P H F F P aF aP X c X c b b

Parallel Lines aH X e X a1b e c X b D2 1 X c P D1 H F F P aF D1 D2 aP

Parallel Planes n c mH f b aH o H 1 F o b aF a1 o b TL c mF f c m1 n n

Parallel Planes If two planes are parallel, any view showing one of the planes in edge view must also show the other plane as parallel edge view. Parallel edge views prove that planes are parallel.

Lines parallel to planes Planes parallel to lines If two lines are parallel, any plane containing one of the lines is parallel to the other line. A line may be drawn parallel to a plane by making it parallel to any line in the plane.

Lines parallel to planes Planes parallel to lines y r m x q p x q m o y p r

Chapter 10 Perpendicularity

Perpendicular Lines If a line is perpendicular to a plane, it is perpendicular to every line in the Plane. e y1 g 90 90 y f j x1 x Perpendicular lines are not necessarily intersecting lines and they do not necessarily Lie in the same plane.

Perpendicular Lines If two lines are perpendicular, they appear perpendicular in any view showing at least one of the lines in true length. If two lines appear perpendicular in a view, they are actually perpendicular in space if at least one of the lines is true length in the same view.

Perpendicular Lines n H 1 s m o n s H o F o TL n s m m

Plane Perpendicular to Line Two-View Method A plane is perpendicular to a line if the plane contains two intersecting lines each of which is perpendicular to the given line.

Plane Perpendicular to Line Two-View Method y h TL z x f H F x h z TL z f y xf EV F h TL 1 y

Plane Perpendicular to Line Auxiliary-View Method k z x H h F x z k z y x EV F k h TL 1 y

Line Perpendicular to Plane Two-View Method A line perpendicular to a plane is perpendicular to all lines in the plane.

Line Perpendicular to Plane Two-View Method f TL o m k H F o a TL k m h f k n

Line Perpendicular to Plane Auxiliary-View Method TL EV n m a k h TL o o m H k F o a k m h k n

Common Perpendicular Point View Method The shortest distance from a point to a line is measured along the perpendicular from the point to a line. The shortest distance between two skew lines is measured by a line perpendicular to each of them.

Common Perpendicular Point View Method b e 1 2 a e H e ab x b F TL a c a e c c b

Common Perpendicular Point View Method b x e 1 y 2 a e x TL H e ab x x b F TL a c y a x e c y c b

Common Perpendicular Plane Method Another method to find the shortest distance between skew lines, specially when the perpendicular view are not required.

Common Perpendicular plane Method 1 c x kh Shortest Distance b k EV TL c h b e e a H a F a e h k c b

Shortest line at specified Grade connecting Two Skew Lines 1 c h TL p c e EV b x ph b a e H Shortest Horizontal Distance F b h a c h p a e

Shortest line at specified Grade connecting Two Skew Lines 1 c h TL p c e EV b x ph b a 100 e H F b h a c 15 h p a e

Projection of line on a Plane The projection of a point on a plane is the point in which a perpendicular from the point to the plane pierces the plane.

Projection of line on a Plane m m 1 ap ap TL b b n n a bp bp h h b TL o o ev F P o m bp ap n

o 2 1 b k X v 3 a 4 n m k 2 o b X v 1 a 4 3 m n