Download - Komfik_Geometri
![Page 1: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/1.jpg)
Komputer GrafikKuliah 1: Prinsip Dasar Komputer Grafik
Dr. Trias Aditya
![Page 2: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/2.jpg)
Rute• Computer Graphic Pipeline Proses Operasi Komputer Grafik Teknik Geodesi & Geomatika & Komputer Grafik Contoh Aplikasi
• Geometri 2D & Geometri 3D Sistem Koordinat Transformasi Koordinat Proyeksi
![Page 3: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/3.jpg)
Computer Graphics Pipeline
![Page 4: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/4.jpg)
Definisi 101
• Komputer Grafik: mencakup tampilan,manipulasi dan penyimpanan gambar dandata eksperimental untuk penyajianmenarik memanfaatkan komputer
• Komputer grafik terdiri dari komputer,memori dan: monitor berwarna, pirantikeras penunjang pemgolahan data dantampilan serta piranti lunak data grafis
![Page 5: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/5.jpg)
Definisi 101
• Komputer Grafik: mencakup tampilan,manipulasi dan penyimpanan gambar dandata eksperimental untuk penyajianmenarik memanfaatkan komputer
• Komputer grafik terdiri dari komputer,memori dan: monitor berwarna, pirantikeras penunjang pemgolahan data dantampilan serta piranti lunak data grafis
![Page 6: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/6.jpg)
• Proses memproduksi gambar atau citramenggunakan komputer proses menulis program untukmenghasilkan/mengolah grafik gambar yang dihasilkan tertayang dilayar komputer sebagai element grafikdisebut “pixel”
![Page 7: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/7.jpg)
ObjectSpecification
SceneDescription
ModelTransformation
SceneDescription
Clipping &Hidden Surface
Removal
ViewTransformation
View & LightSpecification
Shading Image
Image DisplayTransformation
Output
MODELING
RENDERING
DISPLAY
McConnell 2006
![Page 8: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/8.jpg)
![Page 9: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/9.jpg)
Komputer Grafik untuk TeknikGeodesi & Geomatika
• Ragam AplikasiKomputer Grafik:- 3D object modeliing- animation- visualization
3D surface model
Depth image Triangle mesh Texture image
Textured 3DWireframe model
![Page 10: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/10.jpg)
• intermap_harrier2.avigallery_30.mpeg
![Page 11: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/11.jpg)
Supporting Disciplines
• Computer science (algorithms, datastructures, software engineering, …)
• Mathematics (geometry, numerical, …)• Physics (Optics, mechanics, …)• Psychology (Colour, perception)• Art and design
![Page 12: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/12.jpg)
Applications
• Computer Aided Design (CAD)• Computer Aided Geometric Design
(CAGD)• Entertainment (animation, games, …)• Geographic Information Systems (GIS)• Visualization (Scientific Vis., Inform. Vis.)• Medical Visualization• …
![Page 13: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/13.jpg)
Interactive Computer Graphics
User
Application
Screen
input
image
![Page 14: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/14.jpg)
![Page 15: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/15.jpg)
A detailedpolygonmesh(thank
you, WetaDigital)
![Page 16: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/16.jpg)
Graphics pipeline
User
Screen
Model
Geometry, colour
Map
Display
Edit
![Page 17: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/17.jpg)
Manipulation of geometry andcolor…
Monsters, Inc
![Page 18: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/18.jpg)
Manipulation of geometry andcolor…
![Page 19: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/19.jpg)
Manipulation of geometry andcolor…
![Page 20: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/20.jpg)
Manipulation of geometry andcolor…
![Page 21: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/21.jpg)
Manipulation of geometry andcolor…
![Page 22: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/22.jpg)
Representations in graphics
Vector Graphics• Image is represented by continuous
geometric objects: lines, curves, etc.
Raster Graphics• Image is represented as an rectangular
grid of coloured squares
![Page 23: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/23.jpg)
Vector graphics
• Graphics objects: geometry + colour• Complexity ~ O(number of objects)• Geometric transformation possible without
loss of information (zoom, rotate, …)• Diagrams, schemes, ...• Examples: PowerPoint, CorelDraw, ...
![Page 24: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/24.jpg)
Raster graphics• Generic• Image processing techniques• Geometric Transformation: loss of
information• Complexity ~ O(number of pixels)• Jagged edges, anti-aliasing• Realistic images, textures, ...• Examples: Paint, PhotoShop, ...
![Page 25: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/25.jpg)
Conversion
Vector graphics
Rasterization, Pattern recognition
Scan conversion
Raster graphics
![Page 26: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/26.jpg)
Hardware
• Vector graphics• Raster graphics• Colour lookup table• 3D rendering hardware
![Page 27: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/27.jpg)
I/O Devices
3D Scanner 2D ScannerCCDCharge couple device
![Page 28: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/28.jpg)
Vector Graphics Hardware
V E C T O R
Display Controller
move 10 20
line 20 40
...
char O
char R
Display list
continuous & smoothlines
no filled objects
random scan
refresh speed dependson complexity of thescene
![Page 29: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/29.jpg)
Raster Graphics Hardware
Video Controller
jaggies (stair casing)
filled objects
(anti)aliasing
R A S T E R
refresh speed independent ofscene complexity
pixel
scan conversion
resolution
bit planes
0 0 0 0 0 00 7 7 7 60 7 7 70 0 00 00
Frame buffer
![Page 30: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/30.jpg)
Colour Lookup Table
0 0 0 0 0 00 7 7 7 6
0 7 7 70 0 0
0 00
Frame buffer
0 0 0
102 255 53
255 255 204
255 102 153
102 0 51
RR GG BB
0
1
2
4
7
...
colourindex
CLUT:
pixel = code
True colour:
pixel = R,G,B
![Page 31: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/31.jpg)
![Page 32: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/32.jpg)
Geometri Komputer Grafik
![Page 33: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/33.jpg)
2D geometric modelling
• Coordinates• Transformations
![Page 34: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/34.jpg)
Coordinates
• Point: position on planep = (px , py)x = (x, y)x = (x1 , x2)x = x1 e1 + x2 e2, e1 = (1, 0), e2 = (0, 1)
• Vector: direction and magnitudev = (vx , vy), etc.
x
yp
v
![Page 35: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/35.jpg)
Vector arithmetic
• Addition of two vectors:v + w = (vx + wx , vy + wy)
• Multiplication vector-scalar:v = (vx , vy)
x
yw
v
v+w
x
y
v
2v
![Page 37: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/37.jpg)
Why??
![Page 38: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/38.jpg)
Pendefinisian Sistem Koordinatpenting karena….
• Terdapat banyak sistem koordinat padaproses/operasi komputer grafik: WorldCoordinates, Model Coordinates, ViewCoordinates
• Perlu kerangka referensi untukmemposisikan obyek dalam dunia nyata,model maupun dalam tampilan.
![Page 39: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/39.jpg)
Contoh
• Penggambaran obyek 2D pada komputer:
x
y
x
y
Koordinat Peta
Koordinat Layar
![Page 40: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/40.jpg)
Transformasi Geometri
![Page 41: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/41.jpg)
Transformasi Geometri padaKomputer Grafik
• Transformasi geometri memegang perananpenting di komputer grafik karena : obyekdipindah, diperbesar untuk menghasilkanefek/modifikasi
• Graphics API (Application ProgrammingInterface) melakukan transformasi untukmenempatkan/merepresentasikan obyeksebagai tampilan
• Perlu dilakukan transformasi 3D ke 2D (obyekdengan geometri 3D ke layar 2D)
![Page 42: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/42.jpg)
Transformation Pipeline
• World Transformation– Model coordinates World coordinates
• View Transformation– World coordinates Camera space
• Projection Transformation– Camera space View Plane
• These are a series of matrix multiplications
![Page 43: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/43.jpg)
World Transformation
• Translation• Rotation• Scaling
+x
+z
+y
World origin
World Coordinates
Local model coordinates
Local model coordinates
![Page 44: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/44.jpg)
View Transformation
+x
+z
+y
World origin
World Coordinates
+y
+x
+z
• Cameraposition
• Look vector
![Page 45: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/45.jpg)
• Perbesaran (scaling), Rotation (rotasi) dantranslasi (translation) merupakan jenistransformasi yang penting pada komputergrafik.
![Page 46: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/46.jpg)
Skala
![Page 47: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/47.jpg)
Rotasi
![Page 48: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/48.jpg)
Pemotongan (Shear)
![Page 49: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/49.jpg)
Translasi (Translation)
NOT LINEAR!!!Perkalian matrix tidak dapat digunakan
![Page 50: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/50.jpg)
Homogeneous Coordinates
• Pada komputer grafik, transformasi yangkompleks (menggunakan matrix)diperlukan.
• Dengan pendekatan Kartesian, translasidan rotasi, perbesaran tidak dapatdigabung menjadi satu proses
Xn
Yn=
X
YT *
![Page 51: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/51.jpg)
Koordinat Homogen
• Untuk merepresentasikan transformasisebagai matrix diperlukan sistem koordinatyang mampu mewadahi rotasi, perbesarandan translasi sebagai matrix multiplikasi.
• X,y,z pada koord. Homogen = x/z, y/zpada koord. Kartesian
• (X0, y0) pada koord. Kartesian = (x0,y0,1)pada koord. Homogen = (z.x0, z.y0, z)dimana z # 0
![Page 52: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/52.jpg)
Koord. Homogen
![Page 53: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/53.jpg)
Translasi pada Koord. Homogen
Frank Klawonn, Intro to Computer Graphics
![Page 54: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/54.jpg)
Koordinat Homogen• Kartu grafis pada komputer bisa bekerja
cepat karena komposisi matrix utuktransformasi dapat dirangkai menjadi satu
• urutan transformasi penting!• secara default dari kanan ke kiri
contoh
![Page 55: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/55.jpg)
Aplikasi Transformasi (contoh)
Frank Klawonn, Intro to Computer Graphics
![Page 56: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/56.jpg)
3D coordinate system
• Pada koordinat 2D : semua datar• Untuk merepresentasikan kedalaman
diperlukan koordinat z• 3 sumbu koordinat!!• Semua titik pada ruang 3D dapat
dinyatakan sebagai x,y,z
![Page 57: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/57.jpg)
3D coordinate system
x
y
z
(5,0,0)
(5,5,0)
(5,5,5)
(0,0,0)
(0,0,5)
(0,5,5)
(5,0,5)
(0,5,5)
a
b
c
d
e
fg
h
i
j
k
l
![Page 58: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/58.jpg)
Koordinat Homogen
• Melibatkan tidak hanya koordinat x,y dan ztetapi juga nilai w (matrix 4 x 4)
• Titik 3D (x,y,z) direpresentasikan sebagai(x,y,z,1).
• Jika nilai w tidak 1 (x/w, y/w, z/w, 1)
![Page 59: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/59.jpg)
Terms and definitions
C
x
y
z A
B
A’
B’
screenz extent
x extent
y extent
view volume
centre of projection
![Page 60: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/60.jpg)
2D -> 3D• good news is that we can still use our drawing, shape, line
and point classes.
• bad news is that they are all going to require some (hopefullysmall) alterations to work in 3d.
![Page 61: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/61.jpg)
3D Points
1zyx•Still using homogenous
coordinates
•Need 4 coords )1,,,( zyx
![Page 62: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/62.jpg)
3D Transformations
– Since we now have an additional coordinateto cope with, our transformation matricesmust also expand to cope.
– each transformation matrix with require anextra row and an extra column to become4x4 matrices.
![Page 63: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/63.jpg)
Scaling
x
y
z
1
.
1000000000000
1zyx
SS
S
zyx
z
y
x
![Page 64: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/64.jpg)
Translation
x
y
z
1
.
1000100010001
1zyx
TTT
zyx
z
y
x
![Page 65: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/65.jpg)
Rotation (about axes)• In 2 dimensions, rotation were defined to take place about
some stationary point - the centre of rotation.
• In 3d, all rotation take place about one of the axes.
x
y
z
x
y
z
x
y
z
+ve
+ve
+ve
![Page 66: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/66.jpg)
Rotation (about axes)
x
y
z
=-90°
-90 about x axis
![Page 67: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/67.jpg)
Rotation
• x-axis
• y-axis
• z-axis
1
.
10000cossin0
0sincos00001
1z
yx
z
yx
1
.
10000cos0sin
00100sin0cos
1z
yx
z
yx
1
.
10000100
00cossin00sincos
1z
yx
z
yx
![Page 68: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/68.jpg)
Local Rotation
• translate the object tobe rotated to theorigin, perform therotation and translateit back to where itstarted from
x
y
z
=-90°
![Page 69: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/69.jpg)
Proyeksi
• Representasi object 3D di atas layar 2D• Asal muasalnya dan seringnya digunakan
oleh juru gambar dan arsitek dalammenggambar desain
• Several different projections existBergantung dari arah penglihatanpengamat
![Page 70: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/70.jpg)
Terms and definitions
C
x
y
z A
B
A’
B’
screenz extent
x extent
y extent
view volume
centre of projection
(also called plane)
![Page 71: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/71.jpg)
Macam Proyeksi
![Page 72: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/72.jpg)
…so?
• We need to calculate the screen coordinates(xsys) of each projected point from the 3dcoordinates– (they’re not the same!)
• Usually!
• Fortunately – we can consider this calculationas a transformation.
• Can use matrices to represent the variousways of doing it.
![Page 73: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/73.jpg)
Parallel Projection
1
.
10000000
00100001
10 z
yx
yx
s
s
![Page 74: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/74.jpg)
Perspective Projection
1
.
1/100000000100001
1
0 zyx
ddz
yx
s
s
![Page 75: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/75.jpg)
Perspective Projection
p (x,y,z)
z
x
screen
ps (xs,ys)C
d
By similar triangles:
dx
dzx s
dz
xxs
1
dz
yys
1
![Page 76: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/76.jpg)
Perspective Projection
10
1
1
.
dz
ydz
x
want So….Divide by
dz
1
![Page 77: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/77.jpg)
Perspective Projection
1
.
1/100000000100001
1
0 zyx
ddz
yx
s
s
![Page 78: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/78.jpg)
Oblique Parallel Projections• A parallel projection is one where all the lines that are
parallel in 3d space remain parallel when projected.
• An oblique parallel projection is one where the object inquestion is viewed “from the side” - unlike the (plain)parallel projection.
• The z axis is drawn at some angle () to the x axis
• A number of standard oblique parallelprojections are used in engineering drawing:
– cavalier
– cabinet
– orthogonal
![Page 79: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/79.jpg)
Cavalier Projection
The length of a line onthe screen is equal to itslength in the model.
This causes a distortionby over emphasising thez-axis
![Page 80: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/80.jpg)
Cabinet Projection
The foreshortening of the zaxis is increased to provide amore “realistic” view.
![Page 81: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/81.jpg)
Orthogonal Projection
The orthogonal projection isjust the simple parallelprojection
![Page 82: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/82.jpg)
Oblique Parallel Projections
x
y
(xs,ys)
(0,0,1)P
Consider the point P:
P can be represented in 3Dspace - (0,0,1)
P can be represented in 2D(screen coords) - (xs,ys)
P can be represented in 2Dvia polar coordinates - (,)
![Page 83: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/83.jpg)
Oblique Parallel Projections
• At (0,0,1)xs = cos ys = sin
• Generally– multiply by z and allow for (non-zero) x and y
xs= x + z..cos ys = y + z..sin
![Page 84: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/84.jpg)
Oblique Parallel Projection
x
y
z
(0,0,1)
Cavalier cabinetand orthogonalprojections can allbe specified interms of ()
-or () since
tan = 1/
![Page 85: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/85.jpg)
Oblique Parallel Projection
= 0 – 360Orthogonal projection= 90=0
= 0 – 360Cabinet projection= 63.4=0.5
= 0 - 360Cavalier projection= 45=1
![Page 86: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/86.jpg)
Oblique Parallel Projectionas a matrix
1
.
100000000sin100cos01
10 z
yx
yx
s
s
![Page 87: Komfik_Geometri](https://reader034.vdokumen.com/reader034/viewer/2022051114/55cf9a14550346d033a05c51/html5/thumbnails/87.jpg)
Isometric Projection
1000060cos060sin0010060cos060sin
All 3 axes are equally foreshortened allowingmeasurements along the axes to made with the same scale.