Surface From Circle Rotation

Descriptive geometry, as the elementary one, studies the real world by its abstractions. But Euclid’s geometry of the real world is conjugated to pseudo-Euclidean geometry, and they make a conjugated pair. As a consequence, each real figure is conjugated with some imaginary pattern. This paper apart from some science facts demonstrates the presence of imaginary patterns in geometric constructions, where the imaginary patterns manifest themselves as singularities or as geometrically imaginary points (GIP) in “Real — Imaginary” conjugate pairs. The study is conducted, as a rule, from simple to complex, from particulars to generals. Rotation of a circle around an arbitrary axis generates, in the general case, a quartic surface. Among the quartic surfaces are a circular torus and a sphere as a special case of the torus. The torus is obtained from the circle rotation around an axis lying in the circle plane. If the axis does not intersect the generating circle, then the surface is called an open torus; when the axis intersects the generating circle, then the surface is called a closed torus; when the rotation axis passes through the center of the generating circle, then the surface is a sphere. The open torus is associated with a bagel, and the closed one — with an apple. The torus is a perfect example for the application of two well-known Guldin’s formulas. Next, the imaginary torus support is considered in this paper, at the end of which the sphere and its imaginary sup - port are considered. Imaginary patterns lead to the complex numbers, in regards to which grieved the great J. Steiner, calling them "hieroglyphs of analysis". But imaginary patterns exist apart from analysis formulas — they are the part of geometry. J.V. Poncelet was the first who understood the imaginary points in 1812, being in Russian captivity in Saratov and, what is important, without analysis formulas at all. Computational geometry often shows quantities, large numbers of real figures, because it takes into account the imaginary images too.

Camera Calibration Method Based on Circle Plane Board

We present a camera calibration method based on circle plane board. The centres of circles on plane are regarded as the characteristic points, which are used to implement camera calibration. The proposed calibration is more accurate than many previous calibration algorithm because of the merit of the coordinate of circle centre being obtained from thousand of of edge pionts of ellipse, which is very reliable to image noise caused by edge extraction algorithm. Experiments shows the proposed algorithm can obtain high precise inner parameters, and lens distortion parameters.

Theoretical Analysis on Manufacturing Hypoid Left-Hand Gears by Generating-Line Method

According to the hypoid gear tooth surface forming principle, a generating-line will be formed in round-plane while a cone and its tangent circle plane do pure rolling, and the hypoid gear is cutting according to the motion equation as hypoid gears generating-line. to tools shape. The milling processing equation of the hypoid left-hand gear tooth surface on the right side gear tooth surface and on the left side gear tooth surface.There are a detailed description of the adjusting-tool , cutting out from ends, dividing, cycle cutting the whole process. The above method can realizes hypoid gearwheel right tooth surface processing.

Tool Contact Maps by Rectangular Grid Decomposition

This paper is intended to contribute to ongoing research [1–3] in geometric modeling of the virtual machining. In geometric modeling the tool paths are verified by performing the machining simulations and also the cutter workpiece engagements (CWEs) are extracted. CWE geometry is a key input to force calculations and feed rate scheduling in milling operations. Finding these engagements is challenging due to the complicated and changing intersection geometry between the cutter and the in-process workpiece. This paper presents a discrete model based methodology for extracting CWEs generated during a multi axis machining of free form surfaces using a range of different types of milling tools. In this method the in-process workpiece is represented by a set of z-axis aligned rectangular grids. Each grid is made up of four planes, with their normals aligned with respect to the x and y-axis of the Cartesian coordinate system. In developing the methodology the parametric representations of the automatically programmed tool (APT)-type milling cutters are used. The milling tool surfaces are decomposed into circles. During the material removal process only some portions of those circles which are called the engagement arcs may contact the in-process workpiece. To find the geometric limits of those arcs the concept of the feasible contact surface is utilized. The CWE extraction simulation is performed through intersecting those arcs with the planes of each rectangular grid. Thus the intersection calculations reduce to circle/plane intersections which can be performed analytically for the geometry found on milling cutters. To be used in the force model, the CWE boundaries are mapped from Euclidean 3D space to a parametric space defined by the engagement angle and the depth-of-cut for a given tool geometry. Then using a sort algorithm the neighboring engagements in the same arc level are combined.

On Mohr's Circle

In the construction of Mohr's circle, clockwise shear is taken as positive instead of negative since otherwise it does not lead to consistency between the physical plane and Mohr's circle plane. Alternatively, taking anticlockwise shear positive and to make rotation consistent, the shear stress axis is taken pointing downwards which appears rather odd. A simple way to change the axes has been suggested to eliminate this oddity and make the physical plane and Mohr's circle plane consistent in every respect.