Parallel Projection

2021 ◽  
pp. 109-152
Author(s):  
Michael C. Abrams
Keyword(s):  
Parallel Projection

Use of Mechanisms Marking Centers of Simplexes in Their 2-Dimensional Projections as Axonographs of Multidimensional Spaces

2021 ◽  
Vol 8 (4) ◽  
pp. 13-23
Author(s):  
Sherzod Abdurahmanov
Keyword(s):  
Dimensional Space ◽  
Spatial Location ◽  
Straight Line Segment ◽  
Barycentric Coordinates ◽  
Parallel Projection ◽  
Problem Solution ◽  
Similarity Coefficients ◽  
Translational Movement ◽  
Straight Line ◽  
Multidimensional Objects

A brief historical excursion into the graphics of geometry of multidimensional spaces at the paper beginning clarifies the problem – the necessary to reduce the number of geometric actions performed when depicting multidimensional objects. The problem solution is based on the properties of geometric figures called N- simplexes, whose number of vertices is equal to N + 1, where N expresses their dimensionality. The barycenter (centroid) of the N-simplex is located at the point that divides the straight-line segment connecting the centroid of the (N–1)-simplex contained in it with the opposite vertex by 1: N. This property is preserved in the parallel projection (axonometry) of the simplex on the drawing plane, that allows the solution of the problem of determining the centroid of the simplex in its axonometry to be assigned to a mechanism which is a special Assembly of pantographs (the author's invention) with similarity coefficients 1:1, 1:2, 1:3, 1:4,...1:N. Next, it is established, that the spatial location of a point in N-dimensional space coincides with the centroid of the simplex, whose vertices are located on the point’s N-fold (barycentric) coordinates. In axonometry, the ends of both first pantograph’s links and the ends of only long links of the remaining ones are inserted into points indicating the projections of its barycentric coordinates and the mechanism node, which serves as a determinator, graphically marks the axonometric location of the point defined by its coordinates along the axes х1, х2, х3 … хN.. The translational movement of the support rods independently of each other can approximate or remote the barycentric coordinates of a point relative to the origin of coordinates, thereby assigning the corresponding axonometric places to the simplex barycenter, which changes its shape in accordance with its points’ occupied places in the coordinate axes. This is an axonograph of N-dimensional space, controlled by a numerical program. The last position indicates the possibility for using the equations of multidimensional spaces’ geometric objects given in the corresponding literature for automatic drawing when compiling such programs.

New methods for projecting a 3D object onto a free-form surface

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jingyu Pei ◽  
Xiaoping Wang ◽  
Leen Zhang ◽  
Yu Zhou ◽  
Jinyuan Qian
Keyword(s):  
Approximate Solution ◽  
New Method ◽  
Free Form ◽  
Parallel Projection ◽  
Content Type ◽  
3D Object ◽  
New Methods ◽  
Free Form Surface ◽  
Newton Raphson ◽  
Raphson Iteration

Purpose This paper aims to provide a series of new methods for projecting a three-dimensional (3D) object onto a free-form surface. The projection algorithms presented can be divided into three types, namely, orthogonal, perspective and parallel projection. Design/methodology/approach For parametric surfaces, the computing strategy of the algorithm is to obtain an approximate solution by using a geometric algorithm, then improve the accuracy of the approximate solution using the Newton–Raphson iteration. For perspective projection and parallel projection on an implicit surface, the strategy replaces Newton–Raphson iteration by multi-segment tracing. The implementation takes two mesh objects as an example of calculating an image projected onto parametric and implicit surfaces. Moreover, a comparison is made for orthogonal projections with Hu’s and Liu’s methods. Findings The results show that the new method can solve the 3D objects projection problem in an effective manner. For orthogonal projection, the time taken by the new method is substantially less than that required for Hu’s method. The new method is also more accurate and faster than Liu’s approach, particularly when the 3D object has a large number of points. Originality/value The algorithms presented in this paper can be applied in many industrial applications such as computer aided design, computer graphics and computer vision.

The Study on the Linking-Up of Free-Form Surface Fiber Placement Track Based on Mesh Creating

2008 ◽  
Vol 392-394 ◽  
pp. 682-687 ◽  
Author(s):  
Zhong Xi Shao ◽  
Hong Ya Fu ◽  
De Cai Li
Keyword(s):  
Projective Plane ◽  
Calculation Method ◽  
New Method ◽  
Free Form ◽  
Parallel Projection ◽  
Reference Vector ◽  
Fiber Placement ◽  
Vertex Point ◽  
Free Form Surface ◽  
Pass Through

When using meshing creating method of FP (fiber placement) track, once the track point falls at some vertex point of mesh element, in the meantime the vertex point happens to be shared by several mesh elements, there needs a reasonable calculation method to select next mesh element which the FP track will pass through. Then it comes to the problem on linking of FP tracks. In order to solve it, in this paper, the author puts forward a new method, in which parallel projection theory is used, project need analytical mesh element and FP reference vector to a sound projective plane, on which the mesh element be selected and the FP track be calculated, then the FP track would be projected back to the placement surface. Program using this method realized a reasonable joint at the shared vertex point of meshing elements, which the FP direction has little change, and the mutation of track doesn’t come forth. So, the correctness of the method, which putted forward in this paper, is proved.

Zooming Diamonds: A New Illusion, and a Warning on the Biological Relevance of Parallel Projection

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 119-119
Author(s):  
R Cowie
Keyword(s):  
Vision Research ◽  
Biological Systems ◽  
Human Vision ◽  
Motion Path ◽  
Parallel Projection ◽  
Shape Discrimination ◽  
Elevation Change ◽  
Size Change ◽  
Trade Off ◽  
Fixed Distance

A new illusion is described. Observers watch parallel projections of ‘diamonds’ (rhombi) tumbling in 3-D. The displays are generated by moving a viewpoint round a fixed rhombus on a trajectory with two components—a circle parallel to the plane of the diamond, and a superimposed change in elevation which varies sinusoidally. To a greater or lesser extent, depending on the phase of the sinusoid, elevation change is misinterpreted as ‘zooming’ in and out—though in fact the projection always corresponds to an object at a fixed distance. The illusion was devised to underline the questions surrounding the treatment of parallel projection in biological systems. The standard formulations considered in computational vision preclude the kind of size - distance trade-off that the illusion demonstrates, but they do imply that observers should be able to register the shape of an object from this kind of display. A less familiar formulation, ‘paraperspective projection’, allows size - distance trade-off as in perspective projection, but it suggests the shape of a lamina should be impossible to recover from motion. Stimuli which promote ‘zooming’ do weaken shape discrimination, but the trade-off is incomplete. A possible solution is that human vision picks out size change in a way that is appropriate when either object or motion path is ‘friendly’, but that misleads when awkward combinations occur. Certainly vision research should avoid assuming that the attractively simple consequences associated with standard parallel projection govern the way biological systems operate.

ROUGH MACHINING FOR MULTI-SCULPTURED SURFACES

1999 ◽  
Vol 23 (2) ◽  
pp. 275-286
Author(s):  
A. Vafaeesefat ◽  
H.A. EIMaraghy
Keyword(s):  
Cutting Planes ◽  
Tool Path ◽  
Tool Path Generation ◽  
Parallel Projection ◽  
Geometric Description ◽  
Sculptured Surfaces ◽  
Rough Machining ◽  
Tool Paths ◽  
Rough Milling ◽  
Bounding Surfaces

This paper present a method to generate 3-axis NC programs for rough milling processes. A raster digitizing of the solid volume delimitated by sculptured surfaces to be machined is first created. This is accomplished by using the so-called Z-buffer created from a parallel projection of all surfaces. Conventional rendering software can be used to generate the Z-buffer. This volume is transformed into a 3-D mesh composed of “empty”, “full”, and “mixed” blocks. Machining is preformed from top to bottom in a sequence of horizontal cutting planes. At each level of planar machining, spiral routines are used to generate the tool path. The proposed method is valid for generating tool paths for general cavities bounded by arbitrary surfaces. One of the notable advantages of the proposed method is that the tool path generation is independent from the geometric description of bounding surfaces. An example is used to illustrate the approach and its advantages.

Perceived Rigidity and Nonrigidity in the Kinetic Depth Effect

Perception ◽  
1993 ◽  
Vol 22 (1) ◽  
pp. 23-34 ◽  
Author(s):  
Giorgio Ganis ◽  
Clara Casco ◽  
Sergio Roncato
Keyword(s):  
Angular Displacement ◽  
Three Dimensional ◽  
Horizontal Axis ◽  
Parallel Projection ◽  
Depth Effect ◽  
Shape Information ◽  
Kinetic Depth Effect ◽  
Motion Process ◽  
Kinetic Depth ◽  
Angle Of Rotation

Stroboscopic simulations of three-dimensional rotating rigid structures can be perceived as highly nonrigid. To investigate this nonrigidity effect a sequence of either three (experiment 2 and 3) or thirty six frames (experiment 4) was used, each consisting of a set of dots with location on the horizontal axis corresponding to the parallel projection of a nominally defined helix. Observers were asked to judge the angle of rotation of eighty helices defined by the factorial combination of eight phase (φ) values (ie difference between the sinusoidal path of one dot and its neighbours) and ten different angular displacement values (α). When in each static frame the dots can be organized into curved dotted line (small values of φ), the perceived 3-D helices are highly nonrigid. But when shape information is not available in each static frame (high values of φ), the helices are perceived as rigid and rotation judgement is possible providing that α < 15°. It appears that at small values of φ observers fail to recover the rigid structure of the helices since the input to the structure from the motion process may be distorted.

Size Constancy in Structure from Motion

Perception ◽  
1995 ◽  
Vol 24 (10) ◽  
pp. 1155-1164 ◽  
Author(s):  
Jessica Turner ◽  
Myron L Braunstein
Keyword(s):  
Structure From Motion ◽  
Parallel Projection ◽  
Depth Information ◽  
Size Constancy ◽  
Motion Display ◽  
Perspective Effect ◽  
The Relationship ◽  
Relative Motions

The relative motions of points in a structure-from-motion display involving parallel projection provide depth information in an object-centered framework: differences in velocity do not reflect differences in distance from an eyepoint. In contrast, size constancy is generally regarded to be a perspective effect, based on the relationship between projected size and distance from an eyepoint. Five subjects judged the relative sizes of objects in structure-from-motion scenes. Although the scenes were displayed without perspective, judged size was related to the simulated separation in depth of the objects. These results suggest that relative depths recovered from object-centered information are incorporated into a viewer-centered framework.

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