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Related Concept Videos

The Distance Formula01:20

The Distance Formula

In geometry, measuring the direct distance between two points on a plane is essential in various practical and theoretical applications. Whether in navigation, engineering, or computer graphics, determining the shortest path between two locations involves using the distance formula. This formula is derived from the Pythagorean Theorem, which relates the lengths of the sides of a right triangle. On a coordinate plane, the horizontal and vertical distances between two points serve as the legs of...
Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Lagrange Multipliers: Two Constraints01:28

Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.
Distance Problem01:29

Distance Problem

When an object's velocity changes over time, the total distance traveled can be determined by summing small displacement intervals over short increments. This approach approximates the true distance through numerical summation and the use of integral calculus. An estimate of the total displacement can be obtained by measuring velocity at regular intervals and multiplying each value by the corresponding time step.If a runner accelerates over the first three seconds of a race, speed measurements...
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Area Computation by the Alternative Coordinate Method

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Related Experiment Video

Updated: Jun 22, 2026

Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads
07:58

Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads

Published on: July 25, 2025

A geometric buildup algorithm for the solution of the distance geometry problem using least-squares approximation.

Atilla Sit1, Zhijun Wu, Yaxiang Yuan

  • 1Department of Mathematics, Program on Bioinformatics and Computational Biology, Iowa State University, Ames, IA, USA. atilla@iastate.edu

Bulletin of Mathematical Biology
|June 18, 2009
PubMed
Summary

A new geometric buildup algorithm enhances protein modeling by preventing rounding errors and tolerating distance inaccuracies. This method improves stability and robustness in determining protein structures, even with sparse data.

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Area of Science:

  • Computational biology
  • Structural biology
  • Biophysics

Background:

  • The distance geometry problem is crucial for protein structure determination.
  • Existing geometric buildup algorithms can suffer from rounding errors and sensitivity to inaccurate distance data.
  • Sparse or noisy distance information poses a significant challenge in computational protein modeling.

Purpose of the Study:

  • To introduce a novel geometric buildup algorithm for solving the distance geometry problem in protein modeling.
  • To enhance the accuracy, stability, and robustness of protein structure determination methods.
  • To address the limitations of existing algorithms regarding error propagation and data sparsity.

Main Methods:

  • Development of a new geometric buildup algorithm utilizing all available distances.
  • Implementation of a least-squares approximation for atom positioning.
  • Application of a specialized singular value decomposition (SVD) method for solving distance equations.
  • Testing the algorithm on protein structures with varying distance data quality and availability.

Main Results:

  • The algorithm successfully prevents the accumulation of rounding errors during buildup calculations.
  • It effectively tolerates and minimizes small errors in the input distance data.
  • The SVD-based least-squares approach demonstrates robustness, particularly with sparse distance information.
  • Improved modeling ability, stability, and robustness of the geometric buildup approach were observed.

Conclusions:

  • The proposed algorithm offers a significant advancement in protein modeling by overcoming key limitations of previous methods.
  • It provides a more reliable and accurate approach to determining protein structures from distance constraints.
  • The method shows strong potential for practical applications in structural biology and drug discovery.