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Parametric Modelling and Sheetmetal Design
Published in Ionuţ Gabriel Ghionea, Cristian Ioan Tarbă, Saša Ćuković, CATIA v5, 2023
Ionuţ Gabriel Ghionea, Cristian Ioan Tarbă, Saša Ćuković
The term parametric originates from mathematics (parametric equation) and refers to the use of certain parameters that can be edited to manipulate or alter the result of an equation or system. While today the term is used in reference to computational design systems, there are precedents for these modern systems in the works of architects, one of the first being Antoni Gaudí (1852–1926).
Differentiation of parametric equations
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
Rather than using a single equation to define two variables with respect to one another, parametric equations exist as a set that relates the two variables to one another with respect to a third variable.
Differentiation of parametric equations
Published in John Bird, Bird's Engineering Mathematics, 2021
Rather than using a single equation to define two variables with respect to one another, parametric equations exist as a set that relates the two variables to one another with respect to a third variable. Some curves are easier to describe using a pair of parametric equations. The co-ordinates x and y of the curve are given using a third variable t, such as x= f(t) and y= g(t), where t is referred to as the parameter. Hence, for a given value of t, a point (x, y) is determined. For example, let t be the time and x and y are the positions of a particle; the parametric equations then describe the path of the particle at different times. Parametric equations are useful in defining three-dimensional curves and surfaces, such as determining the velocity or acceleration of a particle following a three-dimensional path. CAD systems use parametric versions of equation. Sometimes in engineering, differentiation of parametric equations is necessary, for example, when determining the radius of curvature of part of the surface when finding the surface tension of a liquid. Knowledge of standard differentials and the function of a function rule from previous chapters are needed to be able to differentiate parametric equations.
Students’ understanding of parametric equations in a collaborative technology-enhanced learning environment
Published in International Journal of Mathematical Education in Science and Technology, 2023
Parametric equations are defined as ‘if and are given as functions , over an interval of -values, then the set of points defined by these equations is a parametric curve. These equations are parametric equations for the curve’ (Thomas et al., 2010, p. 610). The researchers assert that covariational reasoning must be used for explaining parametric equations so that students can produce f(t) and g(t) graphs separately through the use of covariational reasoning and can conceptualize the curve as (x,y) = (f(t),g(t)) in which x and y values could be tracked as t varies. However, students might have difficulties in making connections between the rate of change of parametric equations’ components and the rate of change of y with respect to x (Bishop & John, 2008). For instance, Carlson et al. (2002) state that students usually have difficulties in the rate of change hence they could not reflect such change correctly on the graph. Stalvey (2014) indicates that such difficulties in the rate of change are specifically observed in the context of parametric equations. Additionally, the researcher points out that students have difficulties in drawing a function graph defined parametrically and a parameterization of the curves.
Complexities in university students’ understanding of parametric equations and curves
Published in International Journal of Mathematical Education in Science and Technology, 2022
Vahid Borji, Hassan Alamolhodaei
In Calculus, students describe plane curves by giving as a function of (e.g. ) or as a function of (e.g. ) or by using a relation between and that defines implicitly as a unique function of . After these, students learn parametric equations and curves in which both and are considered in terms of a third variable . The way of drawing and finding derivatives for these equations is somewhat different compared to the above-mentioned functions. It may cause students difficulty in learning this concept. In the concept of parametric equations and curves, students learn new methods to describe graphs in the plane. Instead of describing a curve as the graph of an equation or function, students should learn a more general method of describing a graph as the path of a moving particle with a position which is changing over time. Parametric equations have many applications in mathematics, physics, and engineering to describe the motion of an object and to find its behaviour. However, research on this topic in mathematics education is scarce. In a recent study, Çekmez (2020) explored the effects of an instructional intervention using GeoGebra on the relationship between the graph of a parametric curve and the derivative of its component functions. The results showed his students had difficulties in reasoning about the shape of a parametric curve by applying the signs of the first-order derivatives of its component functions. Although Çekmez (2020) found considerable results associated with students’ difficulties of the relationship between the graph of a parametric curve and the sign of the derivatives of its component functions, we belive calculus students have yet early problems with drawing the graph of a parametric equation and finding the first and the second derivative of parametric equations, even when students have the equations of the component functions (i.e. and ). In the current study, we focused on students’ performance when faced with calculus questions associated with sketching the curve of a parametric equation and identifying the direction of the curve, also computing the first and the second derivative of parametric equations. All of these in the context that students have the equation of the component functions.