Lathe Graphics: Exploring Visual Manipulation in Lathe Graphics through Computer Vision

By Fouad Sabry

What is Lathe Graphics

In 3D computer graphics, a lathed object is a 3D model whose vertex geometry is produced by rotating the points of a spline or other point set around a fixed axis. The lathing may be partial; the amount of rotation is not necessarily a full 360 degrees. The point set providing the initial source data can be thought of as a cross section through the object along a plane containing its axis of radial symmetry.

How you will benefit

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

Chapter 1: Lathe (graphics)

Chapter 2: Bucket argument

Chapter 3: Coriolis force

Chapter 4: Sphere

Chapter 5: Rotation

Chapter 6: Cam

Chapter 7: Right-hand rule

Chapter 8: Metalworking

Chapter 9: Magnus effect

Chapter 10: Surface of revolution

(II) Answering the public top questions about lathe graphics.

(III) Real world examples for the usage of lathe graphics 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 Lathe Graphics.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 4, 2024

Book preview

Chapter 1: Lathe (graphics)

A lathed object is a 3D model whose vertex geometry is formed by rotating the points of a spline or other point set around a fixed axis in 3D computer graphics. Lathing may be incomplete; the amount of rotation need not be a full 360 degrees. The initial source data point set can be viewed as a cross section across the object along a plane containing its axis of radial symmetry.

The lathe derives its name from the fact that, like a true lathe, it generates symmetrical objects around a rotational axis.

Similar to surfaces of revolution are lathes. In contrast, lathes are manufactured by rotating a curve defined by a set of points rather than a function. Notably, this implies that lathes can be formed by rotating closed curves or curves that double back on themselves (such as the aforementioned torus), however a surface of revolution cannot, as these curves cannot be characterized by functions.

{End Chapter 1}

Chapter 2: Bucket argument

The objective of Isaac Newton's rotating bucket argument (also known as Newton's bucket) was to demonstrate that real rotational motion cannot be characterized as the rotation of a body with respect to its immediate surroundings. It is one of five arguments from the properties, causes, and effects of true motion and rest that support his claim that, in general, true motion and rest cannot be described as specific instances of motion or rest relative to other bodies, but can be defined exclusively in terms of absolute space. In contrast, these experiments provide an operational description of what is meant by absolute rotation and do not purport to answer the question rotation relative to what?

These arguments and a discussion of the distinctions between absolute and relative time, space, place, and motion appear in a scholium at the end of Definitions sections in Book I of Newton's work, The Mathematical Principles of Natural Philosophy (1687) (not to be confused with General Scholium at the end of Book III), which established the foundations of classical mechanics and introduced his law of universal gravitation, which yielded the first quantitatively adequate description of gravitational attraction.

Descartes recognized, however, that there would be a significant difference between a situation in which a body with movable parts and originally at rest with respect to a surrounding ring was accelerated to a certain angular velocity with respect to the ring and a situation in which the surrounding ring were given an opposite acceleration with respect to the central object. Assuming that both the center object and the surrounding ring were totally rigid, the motions would be indistinguishable if only the central object and the surrounding ring were considered. However, if neither the center item nor the surrounding ring were perfectly rigid, pieces of one or both of them would tend to fly away from the axis of rotation.

Descartes, for contingent reasons relating to the Inquisition, described motion as both absolute and relative.

Consequently, when we declare that a body maintains its direction and velocity in space, we are referring to the entire cosmos in a condensed form.

— Ernst Mach; as quoted by Ciufolini and Wheeler: Gravitation and Inertia, p.

387

Newton explains a water-filled bucket (Latin: situla) suspended by a cord. If the cord is firmly wound around itself and the bucket is subsequently released, it begins to spin fast, not only in regard to the experimenter but also with respect to the water it holds. (This condition corresponds to the preceding diagram B.)

Despite the fact that the relative motion is greatest at this moment, the surface of the water remains flat, demonstrating that the water does not tend to move away from the axis of relative motion despite its near to the pail. Eventually, as the string continues to unwind, the surface of the water becomes a concave form as it acquires the relative motion of the spinning bucket. Despite the fact that the water is at rest with respect to the pail, this concave shape indicates that the water is rotating. In other words, the concavity of the water is not caused by the relative motion of the pail and water, contrary to the notion that motions may only be relative and there is no absolute motion. (This scenario corresponds with diagram D.) Perhaps the concavity of water demonstrates rotation relative to something else, such as absolute space? Newton asserts that the genuine and absolute circular motion of water can be determined and measured.

If a vessel, hung by a long cord, is turned so frequently that the cord becomes tightly twisted, then filled with water, and held at rest with the water; then, by the sudden action of another force, it is whirled around in the opposite direction, and while the cord is untwisting itself, the vessel continues this motion for some time, the water's surface will at first be flat, as it was before the vessel began to move, but the vessel will gradually communicate its motion to the water This climb of the water demonstrates its effort to move away from the axis of its motion; and the true and absolute circular motion of the water, which is immediately opposed to the relative, is revealed and can be measured by this effort. Consequently, this endeavor does not depend on any translation of the water relative to surrounding things, nor can true circular motion be described by such translation....; nevertheless, relative motions have no real effect. It is indeed a matter of extreme difficulty to find and effectively discern the true motions of individual bodies from their apparent motions, because the portions of the immobile space in which these motions occur are not observable by our senses.

— Isaac Newton; Principia, Book 1: Scholium

The argument that motion is absolute and not relative is insufficient since it restricts the experiment participants to the bucket and the water, a limitation that has not been demonstrated. In actuality, the concavity of the ocean incorporates gravitational pull, and by inference, so does the Earth. Due to Mach's argument that only relative motion is demonstrated, here is a critique:

Newton's experiment with the rotating water vessel merely demonstrates that the relative rotation of the water with respect to the vessel's sides produces no discernible centrifugal forces, but that such forces are produced by the water's relative rotations with respect to the mass of the earth and other celestial bodies.

— Ernst Mach, as quoted by L.

Bouquiaux in Leibniz, p.

104

Mach's principle discusses the extent to which Mach's theory is incorporated into general relativity; it is generally accepted that general relativity is not totally Machian.

Everyone agrees that the surface of water that is spinning is curved. The explanation for this curvature, however, entails centrifugal force for all observers except for a genuinely stationary observer, who observes that the curvature is consistent with the rate of rotation of the water as observed, without the requirement for additional centrifugal force. Therefore, a stationary frame can be determined without the need to inquire, Stationary relative to what?:

The question according to what frame of reference do the laws of motion apply? was improperly posed. For the laws of motion define a class of reference frames and (in principle) a construction technique for them.

Newton also presented a second thought experiment with the same goal of determining the occurrence of absolute rotation: watching two identical spheres in rotation about their center of gravity while connected by a string. Tension in the string indicates absolute rotation; refer to Rotating spheres.

The rotating bucket experiment is