内容正文:
nullThe Mathematics of Perspective Drawing:
From Vanishing Points to Projective Geometry
Randall Pyke
March 2019
This presentation: www.sfu.ca/~rpyke presentations perspective
(www.sfu.ca/~rpyke/perspective.pdf)
Perspective, from the Latin perspecta, which means ‘to look through’
The Mathematics of Perspective Drawing:
From Vanishing Points to Projective Geometry
Perspective, from the Latin perspecta, which means ‘to look through’
Look through a pane of glass at an object on the other side,
The Mathematics of Perspective Drawing:
From Vanishing Points to Projective Geometry
The image we see traces out a shape on the glass
From: D’Amelio
We would like to ‘trace’ this image onto the window; this creates a 2
dimensional representation (‘rendering’) of the 3 dimensional scene
– a painting!
Different plane, different perspective…
Humans have been making paintings since the beginning of
time. Conceptual, metaphorical, but not realistic.
Cave painting. Libyan desert, 7000 BC
It took them a long time to figure out how to realistically create a 2
dimensional image of the 3 dimensional world (‘realism’).
Even in the 14th Century paintings were not too realistic (however,
they were very conceptual)
Ambrogio Lorenzetti (Italian) 1290 - 1348
Giotto di Bondone (Italian) 1267 - 1337
12th Century, Song Dynasty
In the 15th Century (Renaissance) painters began to understand how to
make realistic paintings by introducing the third dimension into their
renderings (‘realism’).
Raffaello (Raphael) Sanzio da Urbino (Italian) 1483 – 1520
Raphael
Pietro Perugino (Italian) 1452 - 1523
Georges Seurat
One technique; trace the scene onto a translucent paper while
maintaining a fixed point of view.
But how to do this when you don’t have a scene to copy from?
What are the rules?
Tracing a scene on a window makes a realistic painting…
observer
Furthermore, objects may appear ‘distorted’ when traced out on
the window. How to create the right distortion?
Square object
Trapezoidal image
From: Kline
Two Principles of Perspective Drawing:
1. Parallel lines meet at infinity: Vanishing points
2. Objects farther way appear smaller: Diminution of size
But how to do this when you don’t have a scene to copy from?
Tracing a scene on a window makes a realistic painting…
Filippo Brunelleschi (1377 – 1446) was one of the first to discover the
rules of perspective.
He used a mirror to demonstrate the accuracy of his paintings.
Using vanishing points and the diminution (shrinking) of sizes of distant
objects create a sense of depth.
From: D’Amelio
Vanishing points; one technique for
creating perspective.
From: D’Amelio
Photographs, of course, capture perspective accurately.
Are there vanishing points here?
Yes, one in the centre
Several vanishing points.
The horizon: Where the observer’s eye level is
From: D’Amelio
horizon
Vanishing points: Parallel lines appear to converge
(because the distance between them is diminishing with distance)
These two lines are horizontal -
parallel to the observer’s eye level -
and so appear to converge on the
observer’s horizon.
horizon
Vanishing points
All lines in a given direction appear to converge to the same point
horizon
Vanishing points
All lines in a given direction appear to converge to the same point
Vanishing points
horizon
All parallel lines appear to converge
美术七年级下册第十课
授课老师:艺术知侣
第五单元· 我心中的大海
开发海洋
develop the ocean
畅想未来
imagine the future
目录
我们的海洋
科幻画概念
创作思路
创意实践
我们的海洋
01
海洋面积
海洋资源
海洋难题
我们的海洋
约3.61亿平方千米,
占全球总面积的71%。
海洋是地球上最大的生态系统之一,包含了各种生物和资源。
海洋资源
矿产资源:海底富含石油、天然气等资源。
生物资源:海洋生物、植物
能源资源:海洋的潮汐、海浪、海流等动力资源。例如潮汐电站、风能发电等等。
海洋难题
生活垃圾污染
油污染
生物多样性丧失
海平面上升、海洋温度升高、海水酸化等等。
移居海洋
“移居海洋”是人类未来的目标,科学家正在设计描绘前所未有的智能化“人造海岛”、“海上城市”、“海底世界”、“海洋农场”等,这样人类才能更充分地利用太阳能、风能、潮汐能等多种资源。
人造海岛
海上城市
人类现已开发的海洋项目
迪拜水下酒店
玻璃底游览船
海底餐厅
海底博物馆
海洋保护合理开发
人类只探索了海洋的5%,想一想,如果将剩下海洋的95%合理开发,未来的海洋世界会是怎样的呢?
让我们来画一幅海洋科幻画表达我们的想法!
科幻画概念
02
符合科学原理发明
什么是科幻画
科幻画
对未来世界的奇思幻想
用绘画形式表达
判断:
哪张不是科幻画?
✔
✔
✖
✖
违反科学原理
没有科学内容
创作思路
03
1.确定主题内容
2.选择构图视点
3.选择表现技法
创作思路:
4.开始动笔作画
海洋环保:《海洋和谐卫士》
1.确定主题内容
海洋资源:
石油、天然气、海洋生物、潮汐、发电站等等。
海洋工作:
海洋城市、学校、医院、警局、车站等等。
海洋家园:
衣、食、住、行、游玩、海底农场等等。
移居海洋
海洋难题:
环境污染、生物灭绝、温度升高、海水酸化等。
主体突出、造型新颖
科技过程清晰
1.确定主题内容
2.选择构图视点
平视
仰视
俯视
3.选择表现技法
水粉
线描
油画棒
第一步
注意构图,定出主体的位置、大小,先画出大内容,再画小内容。
4.作画步骤
第二步
丰富画面,将造型细节刻画表现。
4.作画步骤
第三步
色彩有冷暖深浅
的变化,先从主体开始上颜色。
4.作画步骤
第四步
丰富画面、增加细节,一幅海洋科幻画完成!
4.作画步骤
4.创意实践
要求:
以《保护海洋.开发海洋》为题,创作或者自定题目创作一幅海洋科幻画。
有科幻元素
有科幻内容场景
内容造型新颖
构图视点独特
海洋能源:《可燃冰加油站》
学生作品
学生作品
美术七年级下册第十课
授课老师:艺术知侣
第五单元· 我心中的大海
谢谢大家
thanks
Lavf56.40.101
Packed by Bilibili XCoder v2.0.2
$$The Art and Science of Depiction
Fredo Durand
MIT- Lab for Computer Science
Linear Perspective
Perspective 2
Perspective 3
Assignments for Monday 30.
• Solso Cognition and the Visual Arts
– Chapter 8 & 9
• Final project
– Firm subject
Perspective 4
Plan of the next sessions
• Drawing and projection
• Denotation
• Tone & color
Perspective 5
Plan
• Intro to perspective
• History of Renaissance perspective
– Early attempts
– Bruneleschi
– Alberti
– De la Francesca
– Leonardo
– Dürer
• Gadgets
• Distortion issues
Perspective 6
Perspective
• Does linear perspective “occur” in nature
– E.g. we “experience” foreshortening
• Perspective or perspectives?
Perspective 7
Pictorial depth cues
• Occlusion
• Size
• Position relative to the horizon
• Convergence of parallels, linear perspective
• Shading, shadow
• Texture gradient
• Aerial perspective
Perspective 8
Introduction to Perspective
• Importance of the single viewpoint
Perspective 9
Linear Perspective
• Image as a window
• St Jerome In His
Study
Antonello da
Messina 1475
Perspective 10
Some accompanying concepts
• Viewpoint
• Foreshortening
• Vanishing point
• Very linked to architectural spaces
Perspective 11
Primary/secondary geometry
• Primary geometry
– Description in 3D object-space
• Secondary geometry
– Description in 2D image-space
Perspective 12
Plan
• Intro to perspective
• History of Renaissance perspective
– Early attempts
– Bruneleschi
– Alberti
– De la Francesca
– Leonardo
– Dürer
• Gadgets
• Distortion issues
Perspective 13
Early attempts
• St John the Baptist Retiring to the Desert
Giovanni di Paolo 1454
Perspective 14
Early attempts
• Giotto, 1297-1299
Perspective 15
Early attempts
• Lorenzetti 1337-1340
Perspective 16
Early attempts
• Lorenzetti 1337-1340
Perspective 17
Giotto’s rules
• Lines and planes above eye level incline
downwards when they move away
• Below eye level upwards when they move away
• Lines to the left incline inwards to the right
Perspective 18
Giotto
• Confirmation of the Rule of St Francis 1325
Perspective 19
Giotto
• Confirmation of the Rule of St Francis 1325
Perspective 20
Early attempts
• Pietro Lorenzetti,
Birth of the
Virgin
1342
Perspective 21
Plan
• Intro to perspective
• History of Renaissance perspective
– Early attempts
– Bruneleschi
– Alberti
– De la Francesca
– Leonardo
– Dürer
• Gadgets
• Distortion issues
Perspective 22
Bruneleschi’s experiments, 1413
• Existing buildings
• Measurement, surveying
• Single viewpoint
• First scientific mimesis
Perspective 23
Bruneleschi’s experiment, 1413
Perspective 24
Bruneleschi’s second experiment
• Cut out
• Note that this is
2-point
perspective
Perspective 25
Limitations
• Existing buildings
• Requires measurement
Perspective 26
Masaccio, The Trinity, 1427
• The oldest perspective
painting
• Probably measurement
• Maybe helped
by Bruneleschi
Perspective 27
Masaccio, The Trinity, 1427
• The oldest perspective
painting
• Santa Maria Novella (Florence)
Perspective 28
Masaccio, The Trinity, 1427
• The oldest perspective
painting
Perspective 29
Plan
• Intro to perspective
• History of Renaissance perspective
– Early attempts
– Bruneleschi
– Alberti
– De la Francesca
– Leonardo
– Dürer
• Gadgets
• Distortion issues
Perspective 30
Leon Battista Alberti, On Painting
• 1435
• Intersections of pyramids
with image surface
• Proportional triangles
• Use side view for
foreshortening
• Check with diagonal
• Use pavement as reference
• Center = prince of rays
Perspective Drawing
and Projective Geometry
Capstone
WeiPing Li
The Renaissance was a cultural movement that
profoundly affected European intellectual life in
the early modern period (15th century).
Beginning in Italy, and spreading to the rest of
Europe by the 16th century, its influence was felt
in literature, philosophy, art, music, politics,
science, religion, and other aspects of intellectual
inquiry. Renaissance scholars employed the
humanist method in study, and searched for
realism and human emotion in art. Humanists
asserted "the genius of man ... the unique and
extraordinary ability of the human mind."
One of the distinguishing features of
Renaissance art was its development of highly
realistic linear perspective. Giotto di Bondone
(1267–1337) is credited with first treating a
painting as a window into space, but it was not
until the demonstrations of architect Filippo
Brunelleschi (1377–1446) and the subsequent
writings of Leon Battista Alberti (1404–1472)
that perspective was formalized as an artistic
technique. The development of perspective was
part of a wider trend towards realism in the arts
and reflecting the general world view of people
connected to it.
Painting not using Perspective Method:
Initial word panel of Psalm
Perspective method not used
Perspective method not used
Perspective method not used
Perspective method not used
Perspective method used:
Masaccio, Trinity
Perspective method used
Perspective method used: Leonardo Da Vinci:
Adoration (unfinished)
Perspective method used
Donatello: St. George and the Dragon
Perspective method used
Donatello: The Feast of Herod
Perspective method used
Donatello: The Feast of Herod
Artists such as Masaccio strove to portray the human
form realistically, developing techniques to render
perspective and light more naturally. Political
philosophers, most famously Niccolò Machiavelli,
sought to describe political life as it really was, and to
improve government on the basis of reason.
The Renaissance could be viewed as an attempt by
intellectuals to study and improve the secular and
worldly, both through the revival of ideas from
antiquity, and through novel approaches to thought.
Perspective Drawing
We think of the canvas as a glass screen. The
light rays from the object pass through the
screen arriving at viewer’s eye to form an
image. Perspective drawing is the section
produced by the intersection of the glass
screen with the light rays.
People
Brunelleschi (1377-1446):discovered/invented
perspective drawing
Brunelleschi ‘s friends were the forerunners:
Masaccio (1401 -1428)
Masolino (1384-1447)
Donatello (1386-1466)
Alberti (1404-1472) published Della Pittura to
popularize method of perspective drawing.
Bronze statue of Fillipo Brunelleschi
Brunelleschi: Santa Maria della Fiore, the Duomo, the
Baptistry
Leon Battista Alberti (1404—1474)
Della Pittura
The method of perspective drawing is developed from
geometric principles in projecting 3-dim object onto 2-
dim plane.
Dürer perspective drawing device
Dürer perspective drawing device
Dürer perspective drawing device
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