Limitations of the point and line source approximations for the determination of geometry factors around

P Karaiskos1, L Sakelliou, P Sandilos

  • 1Physics Department, University of Athens, Greece. pkaraisk@cc.uoa.gr

Medical Physics
|February 5, 2000
PubMed

Related Concept Videos

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...
Accuracy, limits, and approximation01:28

Accuracy, limits, and approximation

Accuracy, limits, and approximations are common in many fields, especially in engineering calculations. These concepts are imperative for ensuring that a given value is as close as possible to its true value.
Accuracy is defined as the closeness of the measured value to the true or actual value. In engineering mechanics, repeated measurements are taken during theoretical or experimental analyses to ensure that the result is precise and accurate.
The accuracy of any solution is based on the...
Lines in Space01:29

Lines in Space

In three-dimensional analytic geometry, a line can be fully described using vector equations when both a point on the line and its direction are known. This approach has practical applications in fields such as engineering and surveying, where precise spatial modeling is essential. For instance, a laser beam from a surveying instrument directed across a construction site can be modeled mathematically as a line using vectors.Let the laser beam originate from a known point P₀, represented by the...
Influence of Earth's Curvature and Atmospheric Refraction on Leveling01:26

Influence of Earth's Curvature and Atmospheric Refraction on Leveling

During leveling, the Earth's curvature and atmospheric refraction introduce deviations in the line of sight from a true horizontal reference. When the line of sight is leveled, it remains perpendicular to the plumb line only at a single point. Beyond this, it deviates due to the Earth’s curvature, represented by the correction C. For a sight distance D, the deviation can be derived using the relationship:This relationship shows that the deviation increases quadratically with distance. Over a...
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...
Polar Coordinates: Problem Solving01:27

Polar Coordinates: Problem Solving

Directional radiation patterns are central to antenna analysis, as they illustrate how signal strength varies with direction. These patterns are often modeled using polar plots, where the radial distance from the origin represents signal intensity at a given angle. A commonly used idealized form is the four-lobed rose curve, which captures the concept of directional beams in a simplified mathematical form.The four-lobed rose curve, described by r = cos⁡(2θ), features four symmetric lobes, each...