APPLICATION OF RULED SURFACES IN FREEFORM AND GEAR METROLOGY

An application of two ruled surfaces (i.e., surfaces generated by a motion of a straight line), a surface of hyperbolic paraboloid and a tangent surface of a cylindrical helix in freeform and gear metrology is introduced in this paper. Both surfaces have been implemented as the main functional figures in several artefacts – metrological calibration standards intended for testing the freeform capabilities of various measuring technologies (e.g., tactile point-to-point measurement and tactile scanning on coordinate measuring machine, optical scanning, computer tomography). Geometrical and mathematical properties of the surface used are summarised, CAD models of all the developed standards are presented and photos of the manufactured standards are shown.

Sweeping surfaces with Natural mate curve of a spatial curve in Euclidean 3-Space

In this paper, we introduce the notion of sweeping surfaces with Natural mate curve of a spatial curve in Euclidean 3-space E3 . We also show that the parametric curves on these surfaces are lines of curvature. Then, we derive the necessary and sufficient condition for the sweeping surface to become a developable ruled surface. In particular, we analyze the necessary and sufficient conditions when the resulting developable surface is a cylinder, cone or tangent surface. Finally, some representative curves are chosen to construct the corresponding developable surfaces which possessing these curves as lines of curvature.

On Certain Surfaces and Curves Associated with the Bennett Linkage

Although the subject of many more investigations than any other skew linkage, the Bennett loop does not yield up its properties easily. Recent work has revealed the axodes, which putatively determine its higher-order kinematics, and some of the derivable geometric features, but the study remains incomplete. The present paper uncovers previously hidden characteristics of significance, namely, the operative regulus in the fixed axode's central tangent surface and the consequent parametric equation of its curve of striction. The approach is extended to an examination of the hyperboloid which contains the loop's joint axes.

The Curve of Striction of the Bennett Loop’s Fixed Axode

In previous work, the algebraic representation of a fixed axode of the Bennett linkage has been revealed as extraordinarily cumbersome. In this sequel we use properties of the ruled surface to determine the central point of a typical generator of the axode and hence its curve of striction as the intersection of two comparatively simple surfaces. Because of its special significance in this case, we also obtain the equation to the central tangent surface. A feature of the investigation is the direct employment of screw vectors in dual format rather than unit line vectors.

Structural analysis of the central and southwestern Sudbury Structure, Southern Province, Canadian Shield

The Sudbury Structure (SS) is an unusual crater structure which acquired its present oval surface shape during northwest-directed ductile thrusting. Lower amphibolite-facies metamorphism accompanied the thrusting which generated a major reverse shear zone. At least 50 km long, the South Range shear zone (SRSZ) transects the South Range of the Sudbury Structure and exhumes a low level of the Sudbury Igneous Complex (SIC). Assuming heterogeneous simple shear in the northwest–southeast vertical plane on northeasterly striking glide surfaces, minimal estimates of net displacement across the SRSZ exceed 8 km. This displacement magnitude and the map pattern of the SIC require the southwest closure of the SS to be steeply plunging, in accord with a hypothetical funnel shape of the SIC. The rocks of the metasedimentary core of the SS are deformed into a family of second-order buckle folds, the tangent surface of which forms an upright open flexure within the first-order structure of the Sudbury synclinorium. The original orientation and bulk rotation of contacts in the SIC are unknown, so its participation in large-scale folding remains uncertain.

The Theory of Contours, and its Applications in Physical Science

1. The word contour is largely used in ordinary language, but its meaning, when so used, is in general very different from its meaning as a scientific term. We speak of the contour of a hill, a cloud, a country, and so on; meaning usually a profile or an outline,—sometimes a particular outline only. Yet, even in this popular use of the word, we have an indication of its more exact significance. Thus, we see that the visible horizon, if we consider it to be a contour line, is the curve in which the earth's surface is met by its tangent-cone the vertex of which is at the observer's eye. The tangent-surface has a constant characteristic; and it is this possession of a distinctive property by all surfaces which give rise to contour lines, which furnishes the reason for the peculiar applicability of the method of contours to physical problems.