Geometry and its Applications in Arts, Nature and Technology 9783030613976, 9783030613983

Table of contents :
Preface
A very personal foreword
Self-sufficient geometry?
A sixth sense
Descriptive geometry and more
Acknowledgments – the network
Table of Contents
Introduction
Where geometry matters
Level of detail
The website
1 An idealized world made of simple elements
1.1 Points, lines, and circles in the drawing plane
Zero-dimensional spheres
Rays without beginning or end
Circles
The most natural curve
Triangles
Congruent and similar triangles
One of the oldest theorems of geometry
The circle steps out of line
Directly congruent figures, twisted into each other
Similar figures, twisted into each other
1.2 Special points inside the triangle
A circle through three points – the circumcenter
The centroid: a point of physical significance
The orthocenter: a point of geometric significance
All three points are guaranteed to lie on a straight line
The incenter – or: a circle on three tangents
Two ways of defining a circle are still missing . . .
▸▸▸ Two nontrivial circle constructions
A circle through nine (!) points in the triangle
The beer mat method
A point of high importance
How can an arbitrary triangle be converted into an equilateral triangle?
▸▸▸ An inventive tessellation
Are there other methods to convert an arbitrary triangle into an equilateral triangle?
The dilemma of angle trisection
Points at infinity
1.3 Elementary building blocks in space
Surfaces without curvature
1.4 Euclidean space
Basic terms and axioms
A pillar of little support
The sum of distances
The Apollonian circles
Harmonic points
All that remains is the product
From the plane into three-dimensional space
Analogies on the sphere
Other distance lines and distance surfaces
1.5 Polarity, duality and inversion
Infinity can mean many things
A close relationship
The principle of duality
An infinite amount and infinitely more
Inversion – not every transformation is linear
An allegedly inexplicable “anomaly”
A mechanism for the realization
Spherical reflection
Purism
▸▸▸ A proof that isn’t: appearances can be deceiving . . .
1.6 Projective and non-Euclidean geometry
Let us discard lengths and angles
The dimensional jump
Ranges of points and pencils of straight lines
Projectivity = likewise linearity
A conic section always remains a conic section
The line at infinity is suddenly “completely normal”
Non-Euclidean geometry
What’s so fascinating about this?
Geometry on the sphere
The curvature of space
2 Projections and shadows: Reduction of the dimension
2.1 The principle of central projection
The points in space are pushed into a plane
Points at infinity are absolutely normal
Points can “disappear”
What happens to straight lines?
A linear mapping
The mapping only works in one direction
From my perspective . . .
A closer examination of objects at infinity
Central projection of planes
2.2 Parallel projection and normal projection by means of restriction
▸▸▸ A simple depiction by means of a horizontal projection
▸▸▸ Reconstruction of a spatial situation based on shadows
▸▸▸ Radar corner reflectors and cat’s eyes
▸▸▸ Mirror optics instead of refractive lenses
2.3 Assigned normal projections
▸▸▸ Which object is it?
▸▸▸ Simple construction of the unfolding
▸▸▸ Assignment problems?
▸▸▸ Shadows on the coordinate-parallel screen planes
▸▸▸ What is broken or dislocated?
▸▸▸ Shadows provide additional information
Determination of visibility for skew straight lines
2.4 Principal lines and the Law of Right Angles
▸▸▸ Non-trivial right angles in a cuboid
▸▸▸ An abundance of right angles . . .
▸▸▸ If and only if
▸▸▸ Luggage X-ray scans – are they normal projections?
▸▸▸ The crescent does not seem to be facing the Sun (Fig. 2.45)
▸▸▸ When exactly do spring and the other seasons begin (Fig. 2.46)?
▸▸▸ Topographic projection
2.5 The difference in technical drawing
▸▸▸ Special additional views
▸▸▸ Virtual sections
▸▸▸ Multiple virtual sections
▸▸▸ Dimensioning
▸▸▸ Sections in construction drawings
3 Polyhedra: multi-faced and multi-sided
3.1 Congruence transformations
3.2 Convex polyhedra
Prisms and pyramids
A series of simple properties
▸▸▸ Shadows of prisms and pyramids
Combination of several convex bodies
▸▸▸ Acceleration of computer programs
Multiple light sources
▸▸▸ Shadow profiles
When do we speak of a polyhedron?
Euler characteristic
3.3 Platonic solids
As simple as it gets
Three objects from equilateral triangles
The last Platonic solid whose beauty defies description
▸▸▸ Dualization of Platonic solids
3.4 Other special classes of polyhedra
Archimedean solids
▸▸▸ Tiling of space
▸▸▸ Different designs of a soccer ball
Rhombic polyhedra
▸▸▸ More polyhedra consisting of congruent rhombuses?
Catalan solids
▸▸▸ Folding
▸▸▸ Foldable flower vases and pots
3.5 Planar sections of prisms and pyramids
▸▸▸ A flexible interactive installation from an Archimedean solid
4 Curved but simple
4.1 Planar and space curves
Basics from planar curve theory
▸▸▸ A circle as an evolute
▸▸▸ Osculating circles of the ellipse
▸▸▸ A curve can be its own evolute
▸▸▸ We steadily rotate the steering wheel
▸▸▸ Envelopes of circles
▸▸▸ Polarizing curves
▸▸▸ Offset curves
▸▸▸ Architectural design using curves
The theory of space curves
▸▸▸ The simplest space curve
Space curves can create surfaces
Contour and outline of a surface
▸▸▸ Why do highlights occur on curved surfaces?
How can the curvature of a surface be measured?
Polygon mesh on a sphere
▸▸▸ A moderately equal distribution of points on the sphere
▸▸▸ Triangulation as a powerful tool
▸▸▸ Geodesic lines – the shortest connection
How can we recognize a developable surface?
Analogous considerations for shadows
4.2 The sphere
▸▸▸ The position of an airplane
▸▸▸ The “spatial Pythagoras”
▸▸▸ Newton’s kissing number problem
Outline and shadow of a sphere
▸▸▸ The tilt of the slender moon crescent – a “result of temperature”?
▸▸▸ Spheres and spherical parts everywhere . . .
The shortest path on a sphere
▸▸▸ From Vienna to Varadero
4.3 Cylinder surfaces
The tried-and-true cylinder of revolution. . .
. . . and general cylinders
4.4 The ellipse as a planar intersection of a cylinder of revolution
A classic proof of spatial geometry
A multitude of applications
Of course, it doesn’t work every time
▸▸▸ Shadows as intersection curves
The process of development. . .
. . . and its inversion (rolling up the development)
5 More about conic sections and developable surfaces
5.1 Cone surfaces
The development of a cone of revolution
▸▸▸ Rolling up onto a cone
▸▸▸ An unbalanced dumbbell rolling on a plane
▸▸▸ A plane rolling on a cone
▸▸▸ Connecting cone between a rectangle and a circle
▸▸▸ The jet of a hairdryer
▸▸▸ The cone according to
A theorem with great technical applicability
5.2 Conic sections
Planar sections of a cone of revolution
What is a degenerate conic section?
▸▸▸ Various definitions of conics
Focal points
Parabolas and catenaries
▸▸▸ Square wheels
How many elements do we need in order to determine a conic section?
When do we get an ambiguous solution?
The inverse paper strip method
Ellipse and hyperbola with axes given by a line element
▸▸▸ Construction of conic sections via spatial interpretation
Construction of the conic sections
▸▸▸ What does an ancient column have to do with an ellipse?
▸▸▸ “Angular distorted” conic sections
▸▸▸ Shadows on a wall
▸▸▸ Parabolic trajectory
▸▸▸ Why are the orbits of planets ellipses?
5.3 General developables
Developables of constant slope
▸▸▸ Function graphs with constant slope
▸▸▸ Star-shaped surfaces with locally constant slope
How is a developable strip created?
▸▸▸ Creating paper strips
▸▸▸ “Development of a developable”
▸▸▸ Approximation of a curved surface through rectangular strips
Let us generate arbitrary developable surfaces
▸▸▸ Developable connecting surfaces
▸▸▸ The connecting developable of two circles
Reflections on a developable surface
▸▸▸ Shiny generators on aluminum cylinders
▸▸▸ Glowing curves
▸▸▸ Light games in the water
▸▸▸ Reflecting compact discs – and curious effects
5.4 Maps and “sphere developments”
The dilemma regarding the development of doubly curved surfaces
▸▸▸ Is it developable after all?
An angleand circle-preserving projection
A practical map
▸▸▸ From Bangalore to Buenos Aires
▸▸▸ The shortest connection
The inversion at a circle from a “higher perspective”
5.5 Reflection in a circle, a sphere, and a cylinder of revolution
▸▸▸ Reflection in a plane
▸▸▸ Reflection in a curved surface
▸▸▸ Reflection in a cylinder of revolution
▸▸▸ Reflection in a sphere
▸▸▸ Conic sections with given focal points that touch a given circle
▸▸▸ Focal lines of a circle
6 Prototypes
6.1 Surfaces of degree two
▸▸▸ A new arm for Eva
Ellipsoids
▸▸▸ The shape of the Earth
▸▸▸ Ellipsoids in dome construction
▸▸▸ Ellipsoids in industrial design
Listening and focusing . . .
▸▸▸ Whispering bowls and odd acoustic phenomena
Paraboloids
▸▸▸ “Receiving dishes”
▸▸▸ Parabolic headlights and sun collectors
Parabolic translation surfaces
Classically beautiful and very useful
Hyperboloids
▸▸▸ The envelope of a rotating cube (Fig. 6.30)
▸▸▸ Result of rows of points on circles in parallel planes
▸▸▸ Stable and light-weight
▸▸▸ Generalization of the generation
▸▸▸ Geodesics on ellipsoids
▸▸▸ Saturn’s shape and its rings
▸▸▸ Cubic circle
▸▸▸ Movements on and around triaxial ellipsoids
▸▸▸ Moving along the generator of a degree-two surface
6.2 Three types of surface points
Curvature of a planar curve
In space, curvature is not at all easy to define!
Principal curvatures
Osculating parabolas lead to an important theorem
▸▸▸ An exciting experiment
▸▸▸ Light edges
▸▸▸ Revolvability with practical consequences
6.3 Surfaces of revolution
▸▸▸ Vase with an ellipse as its finishing part
▸▸▸ Distorted wine barrel
▸▸▸ When geometers take a walk in nature . . .
The torus is a frequent building block of higher degree
Non-trivial sections of the torus
6.4 The torus as a prototype for all other surfaces of revolution
▸▸▸ Contact lenses and astigmatism
Cusps of contours
Intersection of surfaces of revolution
Reverse engineering
▸▸▸ Gauss map
▸▸▸ Automatic detection of developables
6.5 Pipe and channel surfaces
How to invent a new class of surfaces . . .
Pipe surfaces and blending surfaces
▸▸▸ Blending surfaces
▸▸▸ The altitudes of a tetrahedron lead to an aesthetic lamp
7 Further remarkable classes of surfaces
7.1 Ruled surfaces
Conoids
Two important and two beautiful applications
▸▸▸ A spatial version of the theorem of the angle of circumference
7.2 Helical surfaces
The fascination behind the helix
When straight lines or circles undergo a helical motion . . .
▸▸▸ The DNA double helix
Different types through different directions of the straight line
There exists only one type of developable helical surface
It is possible to apply screw motions to circles or spheres
▸▸▸ The secret of DNA
7.3 Different types of spiral surfaces
Helispiral motion
The classical spiral motion
The contour of a spiral surface
The relation to the helix
7.4 Translation surfaces
Translation surfaces as the loci of chord midpoints
Paraboloids as translation surfaces
The helicoid as a translation surface
Other surfaces of revolution that are also translation surfaces
▸▸▸ Regarding rhombuses and huge towers
7.5 Minimal surfaces
The principal curvatures must be of equal magnitude at each point
▸▸▸ Hyperbolic paraboloids are “almost” minimal surfaces.
▸▸▸ Roofs of minimal surfaces
▸▸▸ Sturdy sculptures with lightweight modules (Fig. 7.74)
▸▸▸ -noids
▸▸▸ “Area-minimizing surfaces”
▸▸▸ An algebraic minimal surface and “something similar”
Bending surfaces into each other
▸▸▸ Bending of minimal surfaces
▸▸▸ Bending of nudibranches and algae
8 The endless variety of curved surfaces
8.1 Mathematical surfaces and free-form surfaces
Function graphs
More general surfaces described by mathematical equations
Different types of surfaces
The addition and subtraction of surfaces
Milled surfaces are also “mathematical surfaces”
How can we solve the problem?
Surface theory is helpful in the creation of categories
▸▸▸ Playing with geometric forms
▸▸▸ The shape of a virus
8.2 Interpolating curves and surfaces
▸▸▸ Growth spurts
Direct modeling at the computer
8.3 Bézierand B-spline-curves
Bézier curves
Rational Bézier curves
B-spline curves
Rational B-spline curves (NURBS)
8.4 Bézierand B-spline surfaces
Generalized Bézier surfaces
▸▸▸ Practical construction
▸▸▸ Nutshells made of plastic
8.5 Surface design done differently
An interplay between art and geometry
A different kind of architectural draft
9 Photographic image and individual perception
9.1 The human eye and the pinhole camera
9.2 Different techniques of perspective
Intersection method
▸▸▸ Shadow constructions in perspective
▸▸▸ Shadow of sidelight and general central projection
Measuring points
▸▸▸ Reconstruction from a photo
▸▸▸ Spatial reconstruction with a given image of a cuboid
▸▸▸ Anamorphoses
▸▸▸ Deliberate distortions in computer graphics
The image of a circle
Circle depictions in ancient times
9.3 Other perspective images
Optical illusions
▸▸▸ The Gaussian collineation: Virtual 3D images
▸▸▸ Two different sequences
▸▸▸ Geoglyphs in the desert sand
▸▸▸ Baroque illusions
▸▸▸ Dynamic Sphere – the illusion of a sphere
▸▸▸ We are only measuring angles!
▸▸▸ Star signs
▸▸▸ The spider on the sphere
▸▸▸ The silhouette of a sphere
▸▸▸ Hyperbolas as silhouettes of spheres?
▸▸▸ When silhouettes of spheres are always circles
▸▸▸ “Smoothing” undesirably extreme perspective images
Image on a sphere
▸▸▸ The geometry of the fish eye
9.4 Geometry at the water surface
Adapted eyes
Reflection at the surface
The Snell-Descartes law of refraction
Image raising
Projecting planes
▸▸▸ Reconstruction with two eyes underwater
Submerged
Refraction at curved (spherical) transitional layers
▸▸▸ The optical system of our eyes
▸▸▸ How does insect vision work?
▸▸▸ A pinhole camera is actually enough!
▸▸▸ The small step towards a real camera
10 Kinematics: Geometry in motion
10.1 The pole around which everything revolves
Constrained motion of two points
▸▸▸ Two-point guidance with a bicycle
Two path normals are sufficient
▸▸▸ Two-point guidance with the bicycle (continuation)
Instantaneous velocity
Envelopes
Relative motion
10.2 Different mechanisms
Let us shake hands!
Two-bar linkages
▸▸▸ The chess-playing android
▸▸▸ The joints of articulate animals
▸▸▸ Leonardo’s cranes
The hinge parallelogram
▸▸▸ Pantograph
Antiparallelograms
It does not always have to be a uniform rotation . . .
The crank shaft
▸▸▸ How fast is the piston?
▸▸▸ Sawmill by Leonardo da Vinci
General four-bar linkages (hinge quadrangles)
▸▸▸ Film gripper
▸▸▸ A useful fixing mechanism (“Mole grips” or “Vise-Grips”)
▸▸▸ The front tires have to be rotated asynchronously!
▸▸▸ Cognate linkages
▸▸▸ A sugar bowl and a four-bar linkage
▸▸▸ How to translate a continuous rotation into an intermittent rotation
If the arms could change their length
▸▸▸ A hinge quadrangle with a single variable
▸▸▸ The Stewart platform
10.3 Ellipse motion
▸▸▸ Construction of paper strips
▸▸▸ Ellipses as hypotrochoids
▸▸▸ Gliding or rolling along straight lines
▸▸▸ Revolving a Reuleaux triangle inside a square
▸▸▸ Drilling quadratic holes in practice
▸▸▸ Milling machine for ellipses
▸▸▸ Oval mill by Leonardo
▸▸▸ A compact coupling
▸▸▸ Ellipses produced by a crank shaft gear
▸▸▸ Ellipse compass according to
▸▸▸ Another ellipse compass
General ellipse motion
▸▸▸ Revolution within a triangle or a pentagram
▸▸▸ The world’s first seven-sided coin
10.4 Trochoid motion
▸▸▸ Wankel’s engine
▸▸▸ Planetary motion
▸▸▸ Planetary gears: plenty of angular momentum
▸▸▸ Involute gears
Cycloids (trochoids) at a higher level
11 Spatial motions
11.1 Motions on the sphere
▸▸▸ The star handle in motion (a rotation about a general axis)
▸▸▸ The Cardan shaft
▸▸▸ A sophisticated ball joint
▸▸▸ “Mad House”
Construction on the sphere
11.2 General spatial motion
Two rotations about skew axes
▸▸▸ An earnest effort on a tiny scale
▸▸▸ The Stewart platform in space
▸▸▸ And yet it moves!
▸▸▸ Non-rigid connections
▸▸▸ “As if it were turned by gears”
11.3 What is the position of the Sun?
Does the Sun not rise in the east and set in the west?
First of all, the facts
Let us now simplify several things
Two important auxiliary theorems
▸▸▸ The locus of all possible sun positions
▸▸▸ Determination of the angle between the Earth’s axis and the sunrays
▸▸▸ Sunrise and sunset
▸▸▸ At which angle of elevation does the Sun culminate?
The incidence of light at a given moment
▸▸▸ Where is south?
▸▸▸ A tiny and versatile gem on your necklace
A constant and a varying angular velocity
The equation of time
▸▸▸ Detective games
▸▸▸ Two comparable and yet slightly different photographs
▸▸▸ When are we allowed to measure an angle in a perspective image?
The path of the Moon on the firmament
▸▸▸ The path of the Moon in the solar system
▸▸▸ What is different on the southern hemisphere?
▸▸▸ Multiple circuits around Antarctica
11.4 Minute-precise sundials for identifying the mean time
True and mean time
Simple sundials
The equation of time
The Bernhardt sundial
Cycloid sundial for mean time
The Pilkington-Gibbs sundials
11.5 Kinematics in the animal kingdom
▸▸▸ Inverse kinematics
▸▸▸ Locomotion in mammals and large birds
▸▸▸ Gripping, biting, and shredding
▸▸▸ Kinematic four-link chain in the monitor lizard head
▸▸▸ How to extend the mouth to a long tube?
▸▸▸ Unfolding the wings
12 A variety of filling patterns
12.1 Ornaments and symmetry groups
Frieze ornaments
Plane symmetry groups
12.2 Periodical tilings
Regular
gons as basic building blocks
▸▸▸ From shark skin to flow-optimized architecture
▸▸▸ Tiling with paintings
▸▸▸ Alteration of the sides of a basic building block
Irregular polygons as basic building blocks
Tiling based on an arbitrary n-gon
Tiling with an arbitrary pentagon
Tiling with an arbitrary hexagon
Periodical tilings with multiple basic forms
12.3 Non-periodical tilings
Self-similar aperiodical tilings
Central aperiodical tiling
Quasiperiodical tilings
▸▸▸ Semi-regular pentagons
▸▸▸ Semi-regular dodecahedrons
▸▸▸ Filling the gaps with scutoids
12.4 Non-Euclidean tilings
What is the situation on the sphere?
▸▸▸ From a dodecahedron to an original and easily manufacturable lamp
▸▸▸ Puzzles in space
13 The nature of geometry and the geometry of nature
13.1 Basic geometric forms in nature
Symmetry
Circles as distance curves and path curves
Spheres and other convex basic forms
Minimal and maximal packets
A maximum of food for the offspring. . .
Don’t roll away!
Unrolling and development
13.2 Evolution and geometry
Mutation
Inheritance
The sum is what matters
Phyllotaxis, or: How easily are we deceived?
A premature conclusion
A completely different explanation
The development is not always clearly visible
13.3 Planetary paths and fish swarms
▸▸▸ Geometry as nature’s artistic inspiration
▸▸▸ Organic architecture
13.4 Scaling behavior in nature
13.5 Musical harmony through the eyes of geometry
▸▸▸ Displaying a tonal system on a torus
▸▸▸ The “grid of keys”
▸▸▸ Three-dimensional circle of fifths
▸▸▸ Tonal centers
▸▸▸ Inverted retrograde canons
14 More applications
14.1 Multiple reflections, illusions, and impossibles
▸▸▸ Multiple reflections in two orthogonal mirrors
▸▸▸ Reflection of a water wave
▸▸▸ What exactly happens between two parallel mirrors?
▸▸▸ Analyzing a selfie
▸▸▸ Vision underneath the surface
▸▸▸ Winding to the left or winding to the right?
Illusions and impossibles
▸▸▸ Hexagons and cubes
▸▸▸Zoetropes
▸▸▸The Freemish crate, also called Escher’s cube
▸▸▸The Penrose triangle, also known as the Penrose tribar
▸▸▸Does this work in perspective images, i.e., in photographs?
▸▸▸Perspective illusions
▸▸▸Rolling upwards?
▸▸▸Inside or outside?
14.2 More about inversions in space
Inversion in a sphere
▸▸▸ Inversion of a torus
▸▸▸ Inversion of cylinder and cone of revolution
▸▸▸ An almost extraterrestial figure
▸▸▸ The loxodromes on a sphere
▸▸▸ Inversion of a clothoid
▸▸▸ Sterographic projection of a clothoid
▸▸▸ Inversion in a cylinder (axial involution)
14.3 Voronoi structures
▸▸▸ Iterative Voronoi structures
▸▸▸ Some classic examples in nature
▸▸▸ Voronoi diagrams on the sphere
▸▸▸ Iterated Voronoi diagrams as a tool to find good point distributions
▸▸▸ Voronoi diagrams in 3-space
An alternative algorithm
14.4 Hexagons on a sphere
▸▸▸ A compound eye under the magnifying glass
▸▸▸ How to tile a sphere with hexagons and some pentagons
▸▸▸ How to tile a sphere exclusively with hexagons
▸▸▸ How to cover a surface with squares or circles
14.5 Fractals
▸▸▸ The Koch snowflake
▸▸▸ Plane-filling curves of dimension two
▸▸▸ Soft coral (gorgonia)
▸▸▸ Camouflage in coral environments
▸▸▸ Fractals on spheres and similar surfaces
▸▸▸ The Mandelbrot set
▸▸▸ Julia sets
▸▸▸ Tree fractals
▸▸▸ Ferns
▸▸▸ Fractal expansion
▸▸▸ The Menger sponge
▸▸▸ Sierpinski fractals
▸▸▸ Percolation
A Geometrical freehand drawing
A.1 Normal view vs. oblique view
Why the oblique view is not our ultimate goal
Isometry
The advantage of a normal projection
The engineer’s view – dimetrical and descriptive
The general solution
▸▸▸ Circles with axes parallel to the coordinate axes
A.2 Don’t be afraid of curved surfaces
Unfamiliar at first. . .
. . . but soon highly advantageous!
It does not get any more difficult
▸▸▸ One-sheeted hyperboloid of revolution
▸▸▸ Beam connection
▸▸▸ Pencil tip
▸▸▸ Helical stairs
▸▸▸ Surfaces of revolution
▸▸▸ Blossom of a lily
A.3 Shadows
▸▸▸ Parallel shadows of a cuboid
▸▸▸ Parallel shadows of cylinders and cones
A.4 Perspective sketching
Reducing the level of difficulty
The method of Renaissance artists
The most important terms
Frontal perspective image
▸▸▸ The “square-grid method”
Perspective images with a horizontal viewing axis
Scaffolding
▸▸▸ A helical staircase in a perspective view
Shadows in perspective images
Reflections
Three principal vanishing points
B Geometry and photography
B.1 Focal lengths and viewing angles
Focal lengths and the viewing angle are connected
▸▸▸ Entire lens systems work like one single converging lens
B.2 3D images in photography?
Are photographs central projections?
An impossible photograph
The lens formula
The plane of sharpness scans the object
B.3 Stereoscopy
Optical illusions with one eye closed
Binocular vision
Stereoscopic vision
The brain readjusts images with collinear distortions
A robust illusion
To put everything into perspective. . .
B.4 When should we use which focal length?
A significant difference
Subjective changes in proportion
From one extreme to the other. . .
. . . or: How a broom closet may be turned into a dining hall
Unintended caricatures and passport photographs
Which focal length should we use underwater?
Depth is crucial
Macro photography underwater
B.5 Primary and secondary projection
Ultra-wide-angle images are occasionally necessary!
Cinema screens and portable TVs
Leonardo’s dilemma
B.6 From below or from above?
Either a shift lens . . .
. . . or some geometrical know-how
▸▸▸ An ad showing long female legs
B.7 Collinear distortion of a photo
B.8 The problem with the rolling shutter
▸▸▸ Can you trust photographs?
▸▸▸ When the shadows do not match
▸▸▸ What every “high-speed photographer” needs to know
▸▸▸ The problem lies in the proximity to reality
▸▸▸ The man that shoots faster than his shadow
▸▸▸ An extremely slow rolling shutter
▸▸▸ What does the image of a rotating straight line look like?
▸▸▸ Photo finish
▸▸▸ Possibilities to avoid the rolling shutter effect
Literature
Quoted literature
Additional recommended books
Picture credits
Index

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