Analytical point-form description of the technique for graphical differentiation of a plane curve
DOI:
https://doi.org/10.15587/1729-4061.2025.343387Keywords:
point polynomial, strip of diffprojections, approximation, analytical chord method, drawings analytizationAbstract
This study considers graphic differentiation, in particular, a chord method, as one of the options for graphic differentiation in terms of replacing graphic operations with analytical ones in point form.
Determining the reference point and the center of projection for constructing a strip of differential projection correlates its positions with respect to the values of the derivative of the function, which is graphically represented by a discrete series of points. The reference point, the right differential projection of the first and left differential projection of the second points have the same values in the field of derivatives. However, they do not coincide with the values of the derivatives of the original functions. To establish such a correspondence, the difference between the left and right differential projections of the first point is divided in half and subtracted from the first derivative of the original function – the point polynomial.
Relative to the reference point, parallel to the first link of the accompanying broken line of the discretely represented curve, a straight line is drawn that intersects the abscissa axis at the center of the projection. Finding the reference point and the projection center is carried out analytically in point form without any graphic operations. Rays are drawn from the projection center parallel to one of the links of the accompanying polyline, thus forming a strip of differential projections, within which the values of the angles of inclination of the tangents to the curve at the base points are selected. Discrete derivative values are connected by straight line segments or remain separate points. The resulting derivative values coincide with the analytical values with a deviation of no more than 0.5–1.5 units.
The developed algorithms could be integrated into automated design and engineering analysis systems for effective calculation of derivatives of discretely given curves. In addition, they could serve as the basis for designing computationally productive modules in artificial intelligence and digital data processing systems that work with geometric and discrete information arrays.
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Copyright (c) 2025 Viktor Vereschaga, Kseniia Lysenko, Yevhen Adoniev, Ernest Murtaziiev, Ivan Vereshchaha, Tetiana Volina
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