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Construction of Concrete Pavements
Published in Rajib B. Mallick, Tahar El-Korchi, Pavement Engineering, 2017
Rajib B. Mallick, Tahar El-Korchi
A hand-operated straightedge 10–20 ft (3–6 m) should be used for checking the surface behind the paving equipment. Successive straightedge checks should overlap by one-half the length of the straight edge to help ensure that the tool detects high and low spots in the surface.
The knowledge of knots: an interdisciplinary literature review
Published in Spatial Cognition & Computation, 2019
Paulo E. Santos, Pedro Cabalar, Roberto Casati
Standing somewhere between the cognitive investigation of knots (described above) and automated reasoning about knot-like structures, Freksa, Barkowsky, Falomir and van de Ven (2018) propose the use of strings and pins, instead of compass and straightedge, for problem-solving in Euclidean geometry. This allows computations to be conducted with physical string-like materials. The motivating argument for this investigation is the idea that the spatial properties, that are essential parts of mathematical proofs, are implicit in diagrammatic representations, whereas they must be spelled out in the analytical (nonvisual) derivations. In this sense, reasoning about geometrical/spatial domains is facilitated by the medium where the inferences are conducted (e.g. strings and pins, as opposed to, say, compass and straightedge). It is argued that this metaphor extends the range of constructive solutions, as exemplified with three strings-and-pins constructions that cannot be represented with compass and straightedge: the construction of an ellipse, the solution for the shortest path problem and the angle trisection problem.
Geometric problem solving with strings and pins
Published in Spatial Cognition & Computation, 2019
Christian Freksa, Thomas Barkowsky, Zoe Falomir, Jasper van de Ven
A string and two pins instantiate the concept of a compass. A taut string with two pins, one of which is fixed at a given location, define a circle. Thus, we introduced new metaphors which by construction instantiate variants of the compass and straightedge that are equally capable of performing Euclidean circle and line constructions. We can use the notion taut string with two pins instead of compass and the notion taut string instead of straightedge for all constructions in Euclidean geometry.
A new perspective on teaching the natural exponential to engineering students
Published in International Journal of Mathematical Education in Science and Technology, 2022
Mukhtar Ullah, Muhammad Naveed Aman, Olaf Wolkenhauer, Jamshed Iqbal
Experience with curves shows that the linear trend of a curve at a point is reasonably captured by a tangent line to the curve at that point. The ‘linear trend’ answers two questions at a point: (i) what is the ordinate of the curve? and (ii) how is the curve directed? The tangent to the curve at a point is unique among all the lines passing through the point in that it is directed along the curve by ‘just touching’ it. Procedures to draw tangents to a circle (and other simple curves) are taught in basic geometry courses. One such example is a compass-and-straightedge construction of a tangent to a circle illustrated in Figure 1. To construct a line t tangent to a circle at a point T on the circle, a line a is drawn from the centre O through T and a line perpendicular to a through T (https://en.wikipedia.org/wiki/Tangent_lines_to_circles) is drawn. Neither the informal description ‘just touching’, nor any geometrical procedure helps understand the relationship between the tangent and the derivative. Students could answer questions about the conditions for a line to be tangent to a curve at a point. Curves of polynomials are familiar to freshman students. As our first example, consider a curve represented by a quadratic polynomial We are seeking the linear trend of the curve near x = r, i.e. a linear polynomial whose graph is a line that passes through, and has a slope m that truly represents the direction of the curve at, the point . Towards that end, let us express in powers of and examine the quadratic remainder Following observations can be made: as the line passes through regardless of the slope m.If , the has a single root at x = r, that is, the line intersects the curve once at and once elsewhere. With intersections at different points, the slope m cannot represent the direction of the curve.If , has a double root at x = r, i.e. the line intersects the curve twice at . With repeated intersections at the point, the slope truly represents the direction of the curve at the point.