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Quadratic models are mathematical representations used to describe relationships in which the rate of change changes at a constant rate. These models appear in a wide variety of natural and engineered systems, especially those involving motion, forces, and optimization. One common application is analyzing the vertical motion of objects influenced by gravity, such as a ball thrown into the air.In such scenarios, the object's height changes over time in a curved pattern, rising to a maximum point...
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A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
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A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of...
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Trigonometric equations involve one or more trigonometric functions and arise frequently in mathematical modeling. These equations may be either identities, which are valid for all values of the variable, or conditional equations, which hold true only for specific values. The process of solving trigonometric equations typically involves both algebraic techniques and the use of fundamental properties of trigonometric functions.Some trigonometric equations resemble standard algebraic forms and...
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Double-angle and half-angle trigonometric identities are derived from the fundamental sum and difference formulas and serve as essential tools for simplifying expressions, solving equations, and evaluating integrals. These identities reduce the complexity of trigonometric functions by relating functions of a multiple or fractional angle to functions of a single angle. Their applications extend across mathematics, physics, and engineering, particularly in Fourier analysis, wave mechanics, and...
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Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
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A quadratic trigonometric spline for curve modeling.

Shamaila Samreen1,2, Muhammad Sarfraz3, Malik Zawwar Hussain2

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A new Quadratic Trigonometric Spline (QTS) offers C2 smoothness and superior geometric properties compared to traditional cubic polynomial splines (CPS). This computationally efficient curve modeling technique enhances applications in science, engineering, and design.

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Area of Science:

  • Computer-Aided Design (CAD)
  • Geometric Modeling
  • Scientific Computing

Background:

  • Traditional cubic polynomial splines (CPS) are widely used but have limitations.
  • Need for advanced spline methods with improved geometric and computational properties.

Purpose of the Study:

  • Introduce a novel curve modeling technique using piecewise quadratic trigonometric functions.
  • Establish a Quadratic Trigonometric Spline (QTS) as a viable alternative to CPS.

Main Methods:

  • Developed a new spline method based on piecewise quadratic trigonometric functions.
  • Ensured C2 continuity for smooth curve generation.
  • Utilized four control points for piecewise curve description.

Main Results:

  • The Quadratic Trigonometric Spline (QTS) achieves C2 smoothness.
  • QTS demonstrates superior geometric properties compared to CPS.
  • Error bounds of order 3 are established for QTS.
  • Comparative analysis shows QTS is a better alternative to CPS.
  • Time analysis confirms QTS is computationally more efficient than CPS.

Conclusions:

  • The proposed Quadratic Trigonometric Spline (QTS) offers significant advantages over traditional cubic polynomial splines (CPS).
  • QTS is a computationally efficient and geometrically favorable method for curve design and control.
  • The technique is applicable across various scientific and engineering disciplines.