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Ref. Descriptive Geometry Metric PARE/LOVING/HILL Fifth edition

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

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

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Perspective Projection Screen Human eye

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Orthographic Projection Screen

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Orthographic Projection Front Horizontal Profil

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Horizontal Front Profil

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Principal Views

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Draw the profile

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Steps to obtain a view 1) Establish the line of sight. 2) Introduce the folding line 3) Transfer distances to the new view 4) Determine visibility and complete the view

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Step 1: Establish the line of sight. 1 Primary Auxiliary Views Step 2: Introduce the folding line Step 3: transfer distances Step 4: determine visibility and complete view k1k1 H 1 y d,h a,e b,f c,g g,h y b,a c,d f,e d h b f c g a e y D1 D2 D1 D2 D

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All views projected from top view has the same height dimension

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Primary Auxiliary Views

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View 1 is an auxiliary view projected from the front View

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

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Edge View of a plane

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Frontal Line

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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.

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Level (Horizontal) Line

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Level Line

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Profile Line

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True Length of an Oblique Line

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Bearing, Slope, and Grade aHaH b 55 aHaH b S55 o E N N Bearing: a term used to describe the direction of a line on the earth’s surface

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aHaH 125 o N Azimuth Bearing b aHaH b N125 o

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

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ahah aFaF N 60 o a1a1 20 o 160 m b b b D1 H F H 1 N

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

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Grade

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Planes

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Points and lines in Planes

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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 a F x m n oFoF n m oPoP

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Solution a F x m n oFoF n m oPoP X Y X Y aPxaPx

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Lines in Planes b c aFaF e b c g e aHaH Complete the front view

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Lines in Planes b c g e aHaH b c aFaF e x x g

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Principal Lines in Planes

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

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Horizontal or Level Lines

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All horizontal lines in the same plane are parallel unless the plane it self is horizontal

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Profile Line

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

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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.

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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

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Solution c aHaH b aFaF b c h h h h f f f f K K 6 mm 5 mm

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

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Pictorial Intersection

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Successive Auxiliary Views

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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.

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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.

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Point View of a Line ahah aFaF 1 b b H F Point View (P.V) T.L. b a2a2 a1,ba1,b 2

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Problem I Find the true clearance between the point O and the line AB. ahah aFaF b b H F T.L. o o b ahah o 1 2 a2,ba2,b o Clearance

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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.

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

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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.

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

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Uses of Auxiliary and additional Views Use Position of line of sight In spaceOn 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 lineParallel to lineParallel 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

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

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a c b a c b f fTL c a b c b a

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

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Piercing Points

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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.

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Piercing point - Auxiliary View Method bHbH c a bFbF c a b 1,c a g e g e e g 1. p p p TL

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Piercing point- Two View Method A piercing point could be found using the given views as follows: ( see the following Fig.) 1) Any convenient cutting plane containing line EG is introduced, it appears EV in a principal view. 2) The line of intersection between the two planes is determined. 3) Since line EG and line both lies in the cutting plane they intersect, locating point P. 4) Since line 1 – 2 also lies in Plane ABC, point P is the required Piercing Point.

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

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bHbH c a bFbF c a g e g e p p Vertical cutting plane N

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Intersection of Planes

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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.

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Intersection of Planes Auxiliary view Method bHbH c a g e x y bFbF c a J k g e J k f f b1b1 c a e k j g x y x y z z

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bHbH c a g e x y bFbF c a J k g e J k bHbH c a e k j g y z y z

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Intersection of Planes Two View - Piercing point Method b a cFcF d eFeF g d ePeP g b a cPcP x x y y

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b a cFcF d eFeF g d ePeP g b a cPcP E.V x x yy L1

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Intersection of Planes Two View - Piercing point Method b a cFcF d eFeF g d ePeP g b a cPcP

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Intersection of Planes Cutting Plane Method H1 H2 P1 P2 Line of intersection b a c m o n

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Intersection of Planes Cutting Plane Method b a cHcH b a cFcF m o nHnH m o nHnH EV of HI P1 EV of H P2 LI P1 P2 LI

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

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

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Angle between Planes

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

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Dihedral Angle Line of Intersection given eHeH LI A g eFeF g e1e1 g B TL LI B A E.V. of A E.V. of B e2ge2g θ

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Dihedral Angle Line of Intersection is NOT given bHbH a c bFbF a c o n m kHkH on m kFkF EV.1 EV x y x y

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Dihedral Angle Line of Intersection is NOT given bHbH a c bFbF a c o n m kHkH on m kFkF x y x y b1b1 a c x y TL o n m k1k1 b2b2 c n m θ X,y

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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

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Dihedral Angle Line of Intersection is NOT given bHbH a c bFbF a c o n m kHkH on m kFkF b2b2 c o n m k1k1 TL b 1,c a EV a o n m k2k2 TS TL Both Planes will Appear EV

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Angle between Oblique Plane and Principal Plane aHaH b c aFaF b c f f TL H F 1 a1a1 b c F θfθf EV of frontal plane Angle between plane and frontal plane

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Angle between Oblique Plane and Principal Plane aHaH b c aFaF b c TL H F EV of Profile plane c b aPaP P f f 1 a1a1 b c θPθP Angle between plane and Profile plane

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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.

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Angle between Oblique Plane and Principal Plane aHaH b c aFaF b c H F Angle between plane and horizontal plane f f 1 aHaH b c θHθH TL

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Parallelism

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

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Parallel Lines aHaH b d H F P c aFaF b d c b d c aPaP

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aHaH b d H F P c aFaF b d c b c d aPaP F

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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.

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Parallel Lines aHaH X c H F P b aFaF b b aPaP F X a 1 b e c X X c P 1

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Parallel Lines aHaH X c H F P b aFaF b b aPaP F X a 1 b e c X X c P 1 e X X e D1 D2

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Parallel Planes mHmH o n mFmF o n aHaH c b aFaF c b f f TL 1 F H o m1m1 n b c a1a1

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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.

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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.

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

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Perpendicularity

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

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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.

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

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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.

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

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

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

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

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

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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.

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

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

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

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

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

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

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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.

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

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X v b a a b m n n m k o ok

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