Curvilinear Perspective: Exploring Depth Perception in Computer Vision

By Fouad Sabry

What is Curvilinear Perspective

Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book La Perspective curviligne, which was translated into English in 1987 as Curvilinear Perspective: From Visual Space to the Constructed Image and published by the University of California Press.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Curvilinear perspective

Chapter 2: Spherical coordinate system

Chapter 3: Tetrahedron

Chapter 4: N-sphere

Chapter 5: Stereographic projection

Chapter 6: Ellipsoid

Chapter 7: Conformal geometry

Chapter 8: 3D projection

Chapter 9: Surface integral

Chapter 10: Volume element

(II) Answering the public top questions about curvilinear perspective.

(III) Real world examples for the usage of curvilinear perspective in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Curvilinear Perspective.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 5, 2024

Book preview

Chapter 1: Curvilinear perspective

Curvilinear perspective, likewise a five-point viewpoint, is a graphic projection used to depict three-dimensional things on two-dimensional surfaces.

It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book La Perspective curviligne, By analogy with a fisheye lens, curvilinear perspective is informally referred to as fisheye viewpoint. In computer animation and motion graphics, it is also known as a miniature planet.

The Arnolfini Portrait (1434) by Flemish Primitive Jan van Eyck contains an early example of approximate five-point curvilinear perspective. Self-Portrait in a Convex Mirror (about 1524) by the mannerist painter Parmigianino and A View of Delft (1652) by the Dutch Golden Age painter Carel Fabritius are later examples.

In 1959, Flocon obtained a copy of Grafiek en tekeningen by M. C. Escher, whose use of curved and curved perspective greatly inspired the theory Flocon and Barre were creating. They began an extensive relationship, during which Escher referred to Flocon as a kindred spirit..

The approach combines both curved perspective lines and an array of straight converging ones to more properly mimic the image on the retina of the eye, which is itself spherical, than the classic linear perspective, which only uses straight lines but is extremely distorted at the edges.

Four, five, or more vanishing points are employed:

In five-point (fisheye) perspective, four vanishing points are set around the circumference of a circle and labeled North, West, South, and East.

Four, or infinite-point perspective is the one that (arguably) most closely resembles the perspective of the human eye, while also being effective for depicting impossible spaces. Just as five point perspective is the curvilinear equivalent of one point perspective, so is four point perspective the equivalent of two point perspective.

Like two-point perspective, this approach can use a vertical line as a horizon line to provide both a worm's eye view and a bird's eye view simultaneously. It employs four or more equally spaced points along a horizon line, all vertical lines are constructed perpendicular to the horizon line, and orthogonals are created using a compass set on a line that passes through each of the four vanishing points at a 90-degree angle.

The distances a and c between the spectator and the wall are greater than b, therefore applying the principle that an object shrinks as its distance from the observer increases, the wall is shrunk and appears warped at its borders.

If a point has the Cartesian coordinates in three dimensions (x,y,z):

{\displaystyle P_{\mathrm {3D} }=(x,y,z)}

Denoting distance from the point to the origin by d = √

x² + y² + z²

, Consequently, the transition of the point to a curvilinear reference system with radius R is

{\displaystyle P_{\mathrm {2D} }=\left({\frac {xR}{d}},{\frac {yR}{d}}\right)}

(if d = 0 and the point is at the origin, its projection is undefined)

This is obtained by first projecting the 3D point onto a sphere with radius R that is centered at the origin, so that an image of the point with coordinates is obtained.

{\displaystyle P_{\mathrm {sphere} }=(x,y,z)*\left({\frac {R}{d}}\right)}

Then, a parallel projection parallel to the z-axis is used to project the point on the sphere onto the paper at z = R, so achieving the result.

{\displaystyle P_{\mathrm {image} }=\left({\frac {xR}{d}},{\frac {yR}{d}},R\right)}

Since it is irrelevant that the paper is resting on the z = R plane, we disregard the z-coordinate of the picture point, yielding

{\displaystyle P_{\mathrm {2D} }=\left({\frac {xR}{d}},{\frac {yR}{d}}\right)=R*\left({\frac {x}{\sqrt {x^{2}+y^{2}+z^{2}}}},{\frac {y}{\sqrt {x^{2}+y^{2}+z^{2}}}}\right)}

Since changing R only amounts to a scaling, Typically, it is characterized as unity, further simplifying the formula to:

{\displaystyle P_{\mathrm {2D} }=\left({\frac {x}{d}},{\frac {y}{d}}\right)=\left({\frac {x}{\sqrt {x^{2}+y^{2}+z^{2}}}},{\frac {y}{\sqrt {x^{2}+y^{2}+z^{2}}}}\right)}

A line that does not pass through the origin is projected onto the sphere as a great circle, which is then projected onto the plane as an ellipse. It is a property of an ellipse that its long axis is a diameter of the bounding circle..

Arrival of Emperor Charles IV at the Basilica of Saint Denis, by Jean Fouquet

Parmigianino, Portrait of Himself in a Convex Mirror

14th century detail of convex mirror in Jan van Eyck's Arnolfini Portrait.

{End Chapter 1}

Chapter 2: Spherical coordinate system

To specify the location of a point in three-dimensional space using a spherical coordinate system, three numbers are used: the radial distance from the origin, the polar angle measured from the zenith, and the azimuthal angle of the orthogonal projection on the plane that passes through the origin and is orthogonal to the zenith. It's like the polar coordinate system, but in three dimensions.

The term radial distance refers to the distance along a circle's radial axis. Colatitude, zenith angle, normal angle, and inclination angle are all names for the polar angle.

When the radius is held constant, the two angular coordinates form a spherical coordinate system.

Different resources and fields may employ different symbols and arrange the coordinates in a different sequence.

This article will use the ISO convention frequently encountered in physics: (r,\theta ,\varphi ) gives the radial distance, polar angle, and the compass's azimuth.

By contrast, several mathematical texts, {\displaystyle (\rho ,\theta ,\varphi )} or (r,\theta ,\varphi ) gives the radial distance, azimuthal angle, the polar angle, switching the meanings of θ and φ.

There are also more idioms in use, for instance, the distance r is from the z-axis, hence, it is essential to double-check the interpretation of the symbols.

Positions are expressed using the language of geographical coordinate systems, Latitude is the metric used to locate objects, longitude, , stature (altitude).

Various celestial coordinate systems exist, each having its own fundamental plane and set of terms for the various angular and linear measurements.

The spherical coordinate systems used in mathematics normally use radians rather than degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system.

Rather than using the polar angle, one might use the elevation angle, which is the angle from the horizontal plane to the positive Z axis, 0 degrees of elevation above the horizon; A negative elevation angle is called a depression angle.

The spherical coordinate system is a more generalization of the polar coordinate system for use in three dimensions. It can be generalized to higher dimensions, at which point it is known as a hyperspherical coordinate system.

A spherical coordinate system is defined by picking an origin point and two reference directions (the zenith and the azimuth) that are perpendicular to each other. A reference plane, including the origin and perpendicular to the zenith, is established by these selections. Then, we can define the spherical coordinates of a point P as follows:

The radial distance, often known as the radius, is the Euclidean distance between two points, say O and P.

Referring to the reference plane, the azimuth (or azimuthal angle) is the signed angle formed by the orthogonal projection of the line segment OP.

The polar angle, or inclination, is the angle formed between the zenith direction and the OP line.

The direction of the azimuth is established by picking a direction that makes a positive turn relative to the zenith. The definition of a coordinate system includes this seemingly arbitrary decision.

To calculate the elevation angle, divide the angle formed between the OP line segment and the reference plane by 2, When the zenith lies on the positive axis.

Equivalently, it is 90 degrees (π/

2

radians) minus the inclination angle.

If the inclination is zero or 180 degrees (π radians), Azimuth can be chosen at will.

Assuming a zero radius, Azimuth and inclination can be chosen at will.

The position vector of a given point P is the vector from O, the origin, to P.

There are a number of various ways in which the three coordinates might be represented, and for the proper sequence in which to compose them.

The use of (r,\theta ,\varphi ) to denote radial distance, inclination (or elevation), and azimuth, respectively, physics routinely use as a method, ISO 80000-2:2019 is the standard that governs this, ISO 31-11 or earlier (1992).

This article, given the foregoing, shall adhere to the ISO standard, {\displaystyle (r,\theta ,\varphi ),} gives the radial distance, polar angle, and the compass's azimuth.

However, some authors (including mathematicians) use ρ for radial distance, φ for inclination (or elevation) and θ for azimuth, noting the distance r is from the z-axis, which offers a natural progression from the conventional notation for polar coordinates.

The azimuth may come before the