Vanishing Point: Exploring the Limits of Vision: Insights from Computer Science

By Fouad Sabry

What is Vanishing Point

A vanishing point is a point on the image plane of a perspective rendering where the two-dimensional perspective projections of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the oculus, or "eye point", from which the image should be viewed for correct perspective geometry. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points.

How you will benefit

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

Chapter 1: Vanishing point

Chapter 2: Perpendicular

Chapter 3: Stereographic projection

Chapter 4: 3D projection

Chapter 5: Oblique projection

Chapter 6: Curvilinear perspective

Chapter 7: Picture plane

Chapter 8: Cross section (geometry)

Chapter 9: Parallel projection

Chapter 10: Axonometry

(II) Answering the public top questions about vanishing point.

(III) Real world examples for the usage of vanishing point 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 Vanishing Point.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 13, 2024

Book preview

Chapter 1: Vanishing point

A vanishing point is the point on the image plane of a perspective rendering where mutually parallel lines in three-dimensional space appear to converge. When parallel lines are perpendicular to a picture plane, the structure is referred to as one-point perspective, and their vanishing point corresponds to the oculus, or eye point, from where the image should be viewed for perfect perspective geometry.

Additionally, the vanishing point is often known as the direction point., as lines with the identical direction vector, say D, have the identical vanishing point.

Mathematically, let q ≡ (x, y, f) be a point lying on the image plane, where f represents the focal length (of the camera associated with the image), and let vq ≡ (x/

h

, y/

h

, f/

h

) be the unit vector associated with q, where h = √

x² + y² + f²

.

If we consider a straight line in space S with the unit vector ns ≡ (nx, ny, nz) and its vanishing point vs, the unit vector associated with vs is equal to ns, assuming both points are directed toward the image plane.

When the image plane is parallel to two world-coordinate axes, images of lines parallel to the axis that this image plane bisects will converge at a single vanishing point. Since they are perpendicular to the image plane, lines parallel to the other two axes will not create vanishing points. This is a single-point viewpoint. When the image plane intersects two world-coordinate axes, parallel lines generate two vanishing points in the picture plane. This is known as dual perspective. In three-point perspective, because the image plane overlaps the x, y, and z axes, lines parallel to these axes intersect, resulting in three distinct vanishing points.

The vanishing point theorem is the most important theorem in perspective science.

It says that the image in a picture plane π of a line L in space, not parallel to the illustration, is determined by its intersection with π and its vanishing point.

Certain authors have employed the phrase, a line's image contains its vanishing point.

Guidobaldo del Monte provided numerous confirmations, and Humphry Ditton called the result the main and Great Proposition.: 244–6 She notes, regarding projective geometry, The vanishing point is the image of L's related infinity point, As the line of sight from O to the vanishing point is parallel to L.

As a vanishing point emerges from a line, so a vanishing line originates in a plane α that is not parallel to the picture π.

Considering the eyepoint O, and β the plane parallel to α and lying on O, then the vanishing line of α is β ∩ π.

For example, when α is the ground plane and β is the horizon plane, then the vanishing line of α is the horizon line β ∩ π.

To put it concisely, the line of no return of a plane, say α, obtained by intersecting the picture plane with a second plane, say β, parallel to the plane of interest (α), traveling past the middle of the camera.

For different sets of lines parallel to this plane α, On this vanishing line will be their respective vanishing points.

The horizon line is a theoretical line that corresponds to the observer's eye level.

If the object is below the horizon, it cannot be seen, Its lines incline toward the horizon.

If the object is above the ground, slope downward.

1.

Projections of two sets of parallel lines lying in some plane πA appear to converge, i.e.

The vanishing point corresponding to this pair, along a horizon, or vanishing line H formed by the intersection of the image plane with the plane parallel to πA and passing through the pinhole.

Proof: Consider the ground plane π, y equals c, which is, In the interest of clarity, parallel to the picture plane.

Also, consider a line L that lies in the plane π, whose definition is given by the equation ax + bz = d.

Using pinhole perspective projections, The coordinates of a point on L projected on the picture plane are, x′ = f·x/

z

= f·d − bz/

az

y′ = f·y/

z

= f·c/

z

This is the parametric representation of the image L′ of the line L with z as the parameter.

When z → −∞ it stops at the point (x′,y′) = (−fb/

a

,0) on the x′ axis of the image plane.

This is the vanishing point corresponding to all parallel lines with slope −b/

a

in the plane π.

All vanishing points associated with different lines with different slopes belonging to plane π will lie on the x′ axis, This is the horizon line in this scenario.

2.

Let A, B, and C be three mutually orthogonal straight lines in space and vA ≡ (xA, yA, f), vB ≡ (xB, yB, f), vC ≡ (xC, yC, f) be the three corresponding vanishing points respectively.

If we know the coordinates of one of these points, we may determine its location, say vA, in addition to the orientation of a straight line on the picture plane, which traverses a second point, say vB, we can compute the coordinates of both vB and vC

A curvilinear perspective is represented by a drawing with four or five vanishing points. In 5-point perspective, vanishing points are mapped into a circle, with four vanishing points at the cardinal directions North, West, South, and East and one at the circle's center.

A drawing with vanishing points placed outside the painting