Descriptive Geometry 1 by Pál Ledneczki Ph.D.

Table of contents 1. Multi-view representation 2. Shadow constructions 3. Intersection problems 4. Metrical problems 5. Axonometry 6. Representation of circle 7. Perspective Introduction

About the purposes of studying Descriptive Geometry: 1. Methods and “means” for solving 3D geometrical construction problems. In this sense Descriptive Geometry is a branch of Geometry.

2. 2D representation of 3D technical object, i.e. basics of Technical Drawing, “instrument” in technical communication.

What is Descriptive Geometry? „One simply takes two planes at right angles to each other, one vertical and the other horizontal then projects the figure to be represented orthogonally on these planes, the projections of all edges and vertices being clearly indicated. The projection on the vertical plane is known as ttehe „eeelevat atoion”, t he ot he r p roject i on i s called „ttepahe plan”. Finaay,telly, the ve etcapaesodedrtical plane is folded about the line of intersection of the two planes until it also is horizontal. This puts on one flat sheet of paper what we ordinarily visualize in 3D”. (A History of Mathematics by Carl B. Boyer, John Wiley & Sons, New York, 1991) Gaspard Monge (1746 ― 1818) was sworn not to divulge the aboveabove method and for 15 years, it was a jealously guarded military secret. Only in 1794, he was allowed to teach it in public at the Ecole Normale, Paris where Lagrange was among the auditors. „With his application of analysis to geometry, this devil of a man will make himself immortal”, exclaimed Lagrange. R. Parthasarathy http:// en.w ikipedi a.org/ /w iki/Gaspar d_ Monge

Descriptive Geometry 1 Introduction 2 About Descriptive Geometry 1

Methodology

Multi-view representation, auxiliary projections

Axonomety

Perspective Types of problems

Incidence and intersection problems, shadow constructions

Metrical constructions

Representation of spatial elements, polyhedrons, circle, sphere, cylinder and cone

Descriptive Geometry 1 Introduction 3 In Descriptive Geometry 1

We shall study

representation of spatial elements and analyze their mutual positions

determine their angles and distances

represent pyramids, prisms, regular polyhedrons,

construct the intersection of polyhedrons with line and plane, intersection of two polyhedrons

construct shadows

cast shadow, self-shadow, projected shadow

the ppprinciples of re presentation and solution of 3D geometrical problems in 2D

Descriptive Geometry 1 Introduction 4 Spatial elements, relations, notation

2 parallel Relations non parallel ï pair of points: determine a distance z,ü perpendicular point and line: lyyging on lying on, passing not lying on → plane, distance through, coincident pair of lines: coplanar D intersecting → angles parallel → distance C non coplanar skew → angle and distance B A l point and plane: lying on k not lying on → distance AB segment A,B line and plane: parallel → distance |AB|= l line A,B intersecting → angle B = k 1 l intersection pair of planes: parallel → distance α = [BCD]plane B,C,D intersecting → angle A α not lying on

Descriptive Geometry 1 Introduction 5 Representation of point

nd 2 quadrant π2 1st quadrant π P” 2 d2 P” P x1,2 P

d1 P’ π1 d1 rd x1,2 3 quadrant

d2 P’ th π1 4 quadrant

In which quadrant or image plane is the point located, why is it special? B” K” C’ N2 D” G” A” H”=J” F” L” B’ d1 N1” X’=X” x1,2 N2” L’ d2 H’ D’ E’=E” C” A’ d = d N1 1 2 K’ F’=G’ J’

Descriptive Geometry 1 Multi-view representation 6 Auxiliary projections

Side view, third image Chain of transformations x2,3 P” d2 P’’’ P” BV

QV C”=D” V x1,2 A”=B” A

V C PV d2

P’ x1,2 Q” DV B’ Fourth image, linked to the first image C’ x4,5 P” IV P’ C DIV d1 Q’ x IV 1,2 P QIV d1 A’ D’

PIV BIV P’ x x 1,4 1,4 AIV

Descriptive Geometry 1 Multi-view representation 7 Representation of Straight Lines, Relative Positions

first principal line second principal line profile line first proj. line second proj. line first coinciding second coinciding v” C” points A”B”=AB B” points L” p ”=M”=N” P” Q” p ” 2 h” A” 1 D” x1,2 x1,2 K” P’ h’ M’ D’ v’ p1’=K’=L’ PPQ=PQ’Q’=PQ A’ B’ p2’ Q’ C’ N’

Intersecting parallel skew intersecting parallel

c” e” k” m” =n” a” g” l” d” b”

c’ a’ g’ k’=l’ m’

b’ n’ d’ e’ first coinciding second coinciding lines lines

Descriptive Geometry 1 Multi-view representation 8 Point and Line

π2 S” π2 p P” N” p P” N” N S K” N R” M” l” l” K P M P R Q” Q” l l Q L” Q L” x1,2 x1,2

N’ N’ l’ Q’ L l’ Q’ L R’ P’ S’ P’ π1 M’ π1 K’ L’ S” L’ P” R” N” N” P” K” M” l” Q” l” Q” p” p” L” L”

N’ N’ N* l’ M’ l’ K’ Q’ Q’ p’ M* p’ P’ R’ P’ S’ L’ L’ R l M p S l K p

Descriptive Geometry 1 Multi-view representation 9 Tracing Points of a Line

l”

N1 π2 N2

x1,2 l” N2’ N1”

l N2 l’ l Q

N1

x1,2 N2’ N1” l’ Problems: π 1) find the tracing points of principal 1 /profile lines 2) determine lines by means of tracing points

Descriptive Geometry 1 Multi-view representation 10 Representation of Plane

Pair of intersecting lines Pair of parallel lines c” a” m” =n” d”

l” k” b” x1,2 c’ m’ a’ k’=l’ n’ b’ 3” d’ Plane figures 2” A” TiTracing lines

n2 n2 4” B” 1” C“ x121,2 B’ 2’ 3’ n1 C’ 1’ n1 4’ A’ spanned plane spanned plane slanted plane slanted plane

Descriptive Geometry 1 Multi-view representation 11 Line and Plane

3” 2” 3” lying on (incident) intersecting 2” l” l [1234] 4” P” 1” g 1 [1234]=P 2’ l”=g” 3’ 1’ 4” l’ 1” 4’ 2’ g’ a” paralle l a” 3’ a 2 [ab] b” P’ 1’ a’ l’ a’ 4’ b’

Descriptive Geometry 1 Multi-view representation 12 Intersection of Two Planes

3” A” 3” 2” 2” 3 A” 2 P2”

P1” P = |12| 1 [ABC] A 1 4” 1” B” C” P B’ P 2 1 P2= |AC| 1 [1234] 2’ B”

3’ B P1’ C 1” C’ 4 P2’ 3’ 1’ B’1 2’ C’

A’ 1’ 4’ A’ 4’

Descriptive Geometry 1 Multi-view representation 13 Transversal of a Pai r Of Skew Lines Passi ng Through a Given Point

Sketch and algorithm Solution in Monge’s Your solution

A A” a” P” a a” t B” B b P” b” b” P a”

x1,2

P’ B = [Pa] 1 b a’ P’ t = |PB| B’ a’ or b’ t = [Pa] 1 [Pb] b’ a’ A’ a2a , [Pa] = [aa]

Descriptive Geometry 1 Multi-view representation 14 Transversal of a Pair Of Skew Lines Parallel to a Given Direction

Sketch and algorithm Solution in Monge’s Your solution t” X” A a” A”

a” * b a t d ” d d” d” X B B” b” b” d* x1,2

d’ b’ X a a’ d*’ B’ d* X, d*2d d’ t’ B = b 1 [ad*] A’ X’ b’ t B, t 2d* a’

Descriptive Geometry 1 Multi-view representation 15 Auxiliary Projections on Special Purposes 1

True length of a segment Distance of a pair of skew lines

A” D” A” B”

C” B” CV x1,2

x1,2 d B’

D’ B’ AV=BV DV C’ IV A’ B BIV A’ x4,5 x141,4

x1,4 CIV AIV DIV AIV

Descriptive Geometry 1 Multi-view representation 16 Auxiliary Projections on Special Purposes 2

Edge view of a plane: transformation of a Application: find the distance d of the point P plane in projecting plane and the plane [ABC].

π4 ü h, π4 ü π1 B” P”

h” A” π4 C” AIV h” x1,2 IV π2 B h A”

d x3,4 B’ h’ CIV PIV A π x 1 1,2 h’ C’ A’ x1,4 P’ A’

Descriptive Geometry 1 Multi-view representation 17 Auxiliary Projections on Special Purposes 3

Construction of the true shape of a figure lying in a general plane General plane 6 fourth projecting plane 6 fifth principal plane

B”

A” h”

π4 C”

IV h” A x 4,5 x1,2 π2 X4,5 h BIV A”

IV BV X1,4 A AV B’ h’ A π CIV x 1 1,2 h’ V π5 C’ C V x1,4 A A’ A’

Descriptive Geometry 1 Multi-view representation 18 Cast Shadow, Self-shadow, Projected Shadow

Descriptive Geometry 1 Shadow constructions 19 Shadow in Traditional Descriptive Geometry

Riess, C.: Grundzüge der darstellenden Geometrie (Stuttgart : Verl. J. B. Metzleráschen Buchhandlung, 1871) Application of Descriptive Geometry for Construction of Projected Shadow (plate X.)

Romsauer Lajos: Ábrázoló geometria (Budapest : Franklin-Társulat, 1929)

http: //www.c 3. hu /perspe ktiva/ ad atb azi s /

Descriptive Geometry 1 Shadow constructions 20 Shadow in Visualisation

Descriptive Geometry 1 Shadow constructions 21 Shadows - Basics

A B”

f” f B B** A A’ f’ A* N2 f

A* z f’ B f B** A’ B*

A x f A* y f’

N1

Descriptive Geometry 1 Shadow constructions 22 Shadow Properties

1) Our constructions are restricted to parallel lighting.

2) We do not represent transition between dark and light shade.

3) We usually construct three types of shadow: cast shadow on the ground or on the image planes, self-shadow (shade) and projected shadow.

4) Shadow of a point: piercing point of the ray of light passing through the point, in the surface (on ground plane, picture plane etc.)

5) Shadow of a straight line: intersection of the plane passing through the line, parallel to the direction of lighting and the surface (screen).

6) Shadow of a curve: the intersection of cylinder (whose generatrix is the curve, the generators are rays of light) with the surface (screen).

7) Shadow-coinciding points: pair of distinct points, whose shadows coincide.

8) Alongside cast shadow the surface is in self-shadow.

9) In case of equal orientation of a triangle and its shadow, the face of triangle is illuminated.

10) The cast shadow outline is the shadow of the self-shadow outline.

Descriptive Geometry 1 Shadow constructions 23 Cast Shadow, Projected Shadow

3 3** 2

A A

shadow coinciding points

A

C 4 C* 1 4* 1*

Descriptive Geometry 1 Shadow constructions 24 Intersection of Pyramid and Line

auxiliary intersection Find the intersection of line and pyramid parallel and similar to the base h

auxiliary intersection passing through the apex l

n1

Descriptive Geometry 1 Intersection problems 25 Intersection of Polyhedron and Projecting Plane

1” Findhd the intersection of pl ane and pol lhdyhedron

D” B” C” A”

3”

M”

4’

D’

1’ C’

M’ 3’ A’

2’ B’

Descriptive Geometry 1 Intersection problems 26 Intersection of Polyhedron and Plane (auxiliary projection)

B Π2

5

1 4

1” A 2 3 C”

h C

M 1’ Π D 1

C’

Hint: introduce Π4 image plane perpendicular to Π1 and the plane of parallelogram (Π4 h6 x1,4 h’).

Descriptive Geometry 1 Intersection problems 27 Intersection of Pyramid and Plane (Collineation)

C n2 P’ M”

a

P A” D” B” C”

D’

C’ The relation between the base polygon and the polygon M’ of intersection is central-axial collineation.

The axis of collineation is the line of intersection of the A’ base plane and the plane of intersection, the center is the apex of the pyramid.

A pair of corresponding points is the pedal point of a B’ lateral edge and the piercing point of the edge in the Hint: start with plane of intersection. |MA| [n n ] n1 W 1 2 The center is the vertex of the pyramid.

Descriptive Geometry 1 Intersection problems 28 Intersection of Prism and Line

Find the intersection of line and prism auxiliary intersection parallel and congruent to the base h

auxiliary intersection parallel to the generators l

n1

Descriptive Geometry 1 Intersection problems 29 Intersection of Prism and Plane (affinity)

A”

n2

P’ A”D”F”B” E” C” E’ P F’

a D’

The relation between the base polygon and the A’ A’ C’ polygon of intersection is axial affinity. The axis of affinity is the line of intersection of the B’ base plane and the plane of intersection. A pair of corresponding points is the pedal point Hint: start with of a lateral edge and the piercing point of the edge n1 |F F| W [n1n2] in the pl ane of iitntersecti on.

Descriptive Geometry 1 Intersection problems 30 Intersection of a Pair of Solids

M” The intersection of two polyhedrons is a polygon (usually 3D polygon). 3” VII VIII VII VIII The vertices of the polygon of I I intersection are the piercing points of the V III V III eddges of a pol yh ed ron in the faces of the other polyhedron. 2” IV IV The edges of the polygon of intersection VI II II VI are segments of intersection of pairs of 1” IX X IX X faces. X1,2 A” B” E” C” D” M 2 III 3 VII VIII E I V 1 IX X II IV VI A’ X X 2 A B C D E A D’ IX IX VI Sequence: I-VII-III-VIII-V-X-VI-IV-II-IX-I V M’ V VI 1’ I At the visibility, one can think of solids or I VII VIII VII VIII surfaces. III IV III II II The visibility depends on, what we want 3’ B’ IV C’ to represent as a result of set operation: union, intersection or a kind of 2’ difference.

Descriptive Geometry 1 Intersection problems 31 Intersection of a Pair of Solids (your solution)

Algorithm: Introduce auxiliary image plane perpendicular to the horizontal edges of the prism Construct the fourth image Find the piercing points of the edges of pyramid in the faces of the prism Find the piercing points of the edges of prism in the faces of the pyramid Find the right sequence of the vertices of polygon of intersection Draw the polygon of intersection in both images Show the visibility

Descriptive Geometry 1 Intersection problems 32 Basic Metrical Constructions 1

The true length of a segment is the hypotenuse of right triangle. One of the legs is the length of A” an image of the segment, the other leg is the difference of distances from the image plane. d1 B”

A” π2 x1,2

d1 A

B’ d1

l d B” AB 1l Q A’ B Reverse problem: the images of a line, a point of the line and a distance is given. Find x1,2 the images of points of the line whose true B’ distance from the given point is equal to the given distance. (Hint: by using an auxiliary point of the line find the ratio of the true A’ π1 lthlength and the lthlength of ii)mage.)

Descriptive Geometry 1 Metrical problems 33 Basic Metrical Constructions 2

Any plan geometrical construction can be carried out by rotating the plane parallel to an image plane. The relation P” between the image of a plane and the image of the rotated plane is orthogonal axial affinity. The axis is a principal line of the plane. OnOnee rotated point can be found h” by the true distance of the point and the axis. x1,2

π2 h’ P’

P

(P) (P) h

x1,2 Reverse problem: construct the images P’ of a figure, whose rotated image is (P) given. Hint: use inverse affinity and h’ lying on condition. π1

Descriptive Geometry 1 Metrical problems 34 Basic Metrical Constructions 3

The first image of a normal of plane, n’ is n” perpendicular to the first image of the first principal v” line h’ of the plane. h” The second image of a normal of plane, n” is perpendicu lar to the second image of the second principal line v” of the plane.

π2 x1,2

n v v’

P h n’ h’

x 1,2 Reverse problem: construct a plane perpendicular to a given line. P’ v’ Hint: the plane can be determined by n’ h’ means of principal lines. π1

Descriptive Geometry 1 Metrical problems 35 Modeling of 3D Polyhedrons

Construct a cube. One of the faces is given by its center and line of an edge. O” e e” O

Algorithm 1) Construct the square lying in plane [O,e], with the centre O and an edge on e. (Rotation x1,2 - counter-rotation of plane, affinity, inverse affinity, (2).) 2) Construct lines perpendicular to the plane [O,e], passing through the vertices of the square. (Perpendicularity of line and plane, (3).) 3) Measure the length of an edge onto the O’ perpendiculars, chose the proper direction from the two possibilities. (True length of a segment, (1).) e’ 4) Comp le te the figfiure by sh owi ng the vi sibilit y.

Descriptive Geometry 1 Metrical problems 36 Step-by-step Construction

Construction of the square Normal of the plane Measure of distance Visibility

P” n” v” B” X” e” e” d1 A ” d x A” O” h” 1 h” A” O”

x1,2 x1,2 x1,2 x1,2

(P) n’ (B) h’ h’ X’ d (e) 1 A ’ d1 v’ x O’ P’ O’ B’ (A) true length e’ of edge e’ A’ A’

Descriptive Geometry 1 Metrical problems 37 Axonometry

One of the methods of Descriptive z Geometry, used to produce pictorial sketches for visualization. Q Let three axes x, y, z through the origin O be given in the image plane. Measure the coordinates from O onto the three axes such that each coordinate will be P multiplied by the ratios of foreshortenings qx, qy, qz respectively. The point 1 determined by the coordinates is z U considered as the axonometric image of z the point P(x, y, z). O The axonometric system can be U 1 U x y R determined by the points y 1 {O, Ux, Uy, Uz}, the image of the origin x and the units on the axes x, y and z. x P’ According to the Fundamental Theorem of Axonometry the axonometric image of an object is a parallel projection or similar to the parallel In aaooxonometry the left-haaddnded Ca rtesia n sy stem is u sed . projection of the object.

Descriptive Geometry 1 Axonometry 38 Fundamental Theorem of Axonometry (Pohlke)

Ez

Ex Let the sytem {O, Ex, Ey, Ez} be given in the image plane. The figure

{O, Ex, Ey, Ez} can be considered as the parallel projection of a

spatial triplet {O, Ex, Ey, Ez}of three unit segments perpendicular by pairs. The spatial triplet can be O one of two types of orientation apart from translation in the E Ey direction of projection. z

Ex O

Ey

http://mathworld.wolfram.com/PohlkesTheorem.html http://www.math-inf.uni-greifswald.de/mathematik+kunst/kuenstler_pohlke.html

Descriptive Geometry 1 Axonometry 39 Orthogonal Axonomtry

z z Z

Z

O O’

Y

y O X Y X x x y O’ is the orthogonal projection of the origin, X, Y and Z are the piercing points of the axes in the image plane. O’ is the orthocenter of the triangle (tracing triangle) XYZ. In the axonometric sketch the prime ( ’ ) is omitted.

Descriptive Geometry 1 Axonometry 40 Orthogonal Axonometry → Multi-view

The orthogonal axonometric image can be considered as one

of an ordered pair of images in multi-view representation. x a,IV z

IV z Z IV Z IV P P

IV O

O X XIV =YIV Y x IV =y IV y P’ x

Descriptive Geometry 1 Axonometry 41 Oblique (klinogonal) Axonometry

Ez

Ex

O E Ez y

Ex O’

Ey

OO’ is not perpendicular to the picture plane. {qx,,q qy,,q qz}

Descriptive Geometry 1 Axonometry 42 Izometry, Technical Axonometry

Izometry Technical axonometry

z z

8 8 120° 120° 1 x 120°

7 y x

q = q = q =1 q = q =1, q = ½ x y z y x z y

Descriptive Geometry 1 Axonometry 43 Cavalier, Bird’s-eye View, Worm’s-eye View

Frontal Axonometry Bird’s eye view (top view) Military axonometry Worm’s eye view (bottom view) z

z

z x y

x y y x

Image plane: [xz] Image plane: [xy] qx = qy = 1 Image plane: [xy] q = q = 1 if qy = 1 : cavalier axonometry if qz = 1: military axonometry x y

Descriptive Geometry 1 Axonometry 44 Frontal Axonometry, Shadow

A

(A)

Descriptive Geometry 1 Axonometry 45 Military Axonometry

www.xanadu.cz http://www.fotosearch.com/NGF001/57478808/

Descriptive Geometry 1 Axonometry 46 Cast Shadow in Orthogonal Axonometry

A*

f

K* A

Dodecahedron, top view and heights f K

x 12 f’

(A) (O) K’

Descriptive Geometry 1 Axonometry 47 Projected Shadow

I* A*

X* J* K* I H* C* A

B* X J C K H Y* B C D* Y D

D

Descriptive Geometry 1 Axonometry 48 Representation of Circle (Multi-view)

3” 6” v” 10” 1. The major axes lie on first and second principal lines 8” h’ and v” respectively. h” 1” O” 2. The length of major axes 1’2’ and 5”6” is equal to the diameter of circle (true length). 2” 3. The length of a minor axis is constructible from the 7” major axis and a point, as plane geometric 9” construction. (See construction of 8”) 5” 4” x1,2 6” 7’ a 8” a b 2’ 1” 4’ b 10’ 5’ v’ 5” 6’ O’ 4. The lef t and righ i ht extreme poi nts 9 and 10 can be 9’ found as points of ellipse with vertical tangents, by 3’ means of orthogonal axial affinity. 1’ 5. The tangents at the points mentioned above are 8’ parallel to the proper diameters. h’

Descriptive Geometry 1 Representation of circle 49 Representation of Circle (Orthogonal Axonometry)

z 1. The major axes of ellipses are perpendicular to the coordinate axes. 2. The minor axes are coinciding lines with the coordinate axes. 3. The fundamental method of constructions is the orthogonal axial affinity. z Oo zo O

x (y)

O (x)

y y x

(O)

Descriptive Geometry 1 Representation of circle 50 Representation of Circle (Oblique Axonometry)

z 1. The fundamental method of constructions is the oblique axial affinity. 2. The axe s can be con st ructed by Ryt z’ method. The ellipse is determined by a pair of conjugated diameters; find the major and minor axes.

Yo L

o y x Q*

Q M -90° P Y y N O

(Y) (y)

Descriptive Geometry 1 Representation of circle 51 Axonometry vs. Perspective

Descriptive Geometry 1 Perspective 52 Vanishing Point, Vanishing Line

Vt

Vt∞ ishing line ishing line van van

V V vanishing line horizon r l vanishing line horizon V Vl r

A set of parallel lines in the scene is projected onto a set of lines in the image that meet in a common point. This point of intersection is called the vanishing point. A vanishing point can be a finite (real) pp()pgpgppoint or an infinite (ideal) point on the image plane. Vanishing points which lie on the same plane in the scene define a line in the image, the so-called the vanishing line.

Descriptive Geometry 1 Perspective 53 Basics of Perspective with Vertical Image Plane

plane of horizon plane of horizon h C F Principal point F C V image plane rotated ground plane P h image plane (C) l rotated center l ground plane a A=(A) P

Ca a

ground plane F V h (P) P

a A=(A) (()C)

Descriptive Geometry 1 Perspective 54 Perspective Collineation, Perspective Mapping

A perspective collillitineation is dtdetermi ne d V1 V3 v V2 by the center C, axis a and the Q’ l’ vanishing line v. P’ l To the square P’, Q’, R’, S=SS’=S, we can R’ R Q P find the quadrilateral PQRS at the a mapping Π’ Y Π. S’=S When the ground plane is rotated into C the picture plane, the two systems of V1 V3 h V2 points and lines are related by central- (Q) (l) axial collineation. This perspective (P) collineation is determined by the Q l (R) R P rotated center (C), axis a and the a horizon h (vanishing line). S’=S To the square (P), (Q), (R), (S)=S, we can find the quadrilateral PQRS at the

(C) mapping (Π) Y Π.

Descriptive Geometry 1 Perspective 55 Heights in Perspective

True height can be measured in the image plane ng line ii vanish

vanishing line h

true height a

Descriptive Geometry 1 Perspective 56 Representation of Circle (Perspective)

In the ground plane In vertical plane (B) (P) h F V

(O) b B (M) (b) (K) O d P a K M A=(A) Q=(Q) [M] [P]

dist((Q),(P)) = dist((A),(B)) (C) M: midpoint of PQ d K: midpoint of AB, center of the ellipse O: image of center of circle d = dist( (v), a) = dist( (C), h)

()(v) vanihiishing line o f ground pl ane

Descriptive Geometry 1 Perspective 57 Cylinder and Cone in Perspective

Axonometric sketch

h

C h C

a

r a Ca

r

Descriptive Geometry 1 Perspective 58 Shadow in Perspective

shadow vanishing point horizon If’

If light vanishing point http://www.math.ubc.ca/~cass/courses/m309-03a/m309-projects/endersby/Antisolarpoint.html

Descriptive Geometry 1 Perspective 59 Shadow Types in Perspective

f

If

h f’

h If’ h If’

If

Descriptive Geometry 1 Perspective 60 Perspective with Slanting Picture Plane

H H h

C F [C] F

Vz

Vz

Descriptive Geometry 1 Perspective 61 Constructions with Slanting Picture Plane

Rotation of the ground plane Rotation of a vertical line

(l) H h (P) V l H h F [C] F [C] P a a (p) N (C)

p

Vz

Descriptive Geometry 1 Perspective 62 Perspective with Slanting Picture Plane

Descriptive Geometry 1 Perspective 63