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Related Concept Videos

Finding Volume Using Cross-Sectional Area01:24

Finding Volume Using Cross-Sectional Area

For solids whose cross-sectional areas vary in a predictable way, volume can be determined by integrating these areas along an axis perpendicular to the slices. This approach is particularly useful for polyhedral solids, where classical geometric formulas may not be immediately applicable. A tetrahedron provides a clear example of how cross-sectional integration can be applied to a three-dimensional object with continuously changing geometry.Consider a tetrahedron with height h and a base that...
Volumes of Solids of Revolution01:29

Volumes of Solids of Revolution

Volumes of irregularly shaped objects can be systematically determined using the concept of solids of revolution. This approach begins with a region defined by a curve in a two-dimensional plane. When this region is rotated about a fixed line, known as the axis of revolution, it generates a three-dimensional object with rotational symmetry. Such objects frequently arise in mathematical modeling, physics, and engineering applications.When the region being rotated lies directly against the axis...
Calculation of Volume of Solids by Integration01:27

Calculation of Volume of Solids by Integration

Volume calculation often begins with simple geometric solids. For example, the volume of a rectangular box is obtained by multiplying the area of its base by its height. This straightforward approach relies on the fact that the cross-sectional area of the box remains constant throughout its length. Many real-world objects, however, do not have uniform cross-sections, and their volumes cannot be determined using elementary geometric formulas.To address this limitation, the Slicing Method...
Cylinders in Three-Dimensional Space01:28

Cylinders in Three-Dimensional Space

A cylindrical surface is generated when a two-dimensional profile curve is translated along a straight line in three-dimensional space. The translated copies of the curve form a surface composed of parallel rulings, each oriented in the same fixed direction. This construction allows many three-dimensional forms to be described using relatively simple planar equations.In Cartesian coordinates, a cylindrical surface is often recognized by an equation that omits one of the three variables. For...
Real-Life Applications of Multiple Integrals01:18

Real-Life Applications of Multiple Integrals

Multiple integrals provide a powerful mathematical framework for calculating physical quantities distributed throughout two- and three-dimensional regions. One important application is the determination of volume in objects with curved geometries, such as storage tanks, pipes, and reservoirs. Cylindrical coordinates are especially useful for systems with rotational symmetry because they simplify the description of circular and paraboloid-shaped regions.Consider a paraboloid-shaped water tank...
Triple Integrals in Cylindrical Coordinates01:28

Triple Integrals in Cylindrical Coordinates

Cylindrical coordinates describe a point in three-dimensional space using three values: radial distance, angle, and height. The height gives the position above the xy-plane, the radial distance measures how far the point is from the z-axis, and the angle describes the point’s direction from the positive x-axis in the xy-plane. This system is especially useful for regions with circular symmetry because it matches the natural geometry of cylinders, disks, and circular tanks.To calculate volume,...

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Methods for managing 3-dimensional volumes.

Asem Awaad Othman1, Amr Ragab El-Beialy, Sahar Ali Fawzy

  • 1Systems and Biomedical Engineering Department, Cairo University, Cairo, Egypt.

American Journal of Orthodontics and Dentofacial Orthopedics : Official Publication of the American Association of Orthodontists, Its Constituent Societies, and the American Board of Orthodontics
|February 16, 2010
PubMed
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New 3D volumetric technology requires advanced orthodontic tools. This study introduces methods for virtual model creation, landmark localization, and region manipulation for improved treatment planning and analysis.

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Area of Science:

  • Orthodontics
  • Medical Imaging
  • Computer-Aided Design

Background:

  • 3D volumetric technology generates vast amounts of data, necessitating novel diagnostic and treatment planning approaches in orthodontics.
  • Existing methods may not fully leverage the potential of 3D imaging for detailed orthodontic analysis.

Purpose of the Study:

  • To introduce new methods and tools for managing 3D images in orthodontic diagnosis and treatment planning.
  • To facilitate the creation of virtual models and automatic landmark identification from 3D patient data.

Main Methods:

  • Development of tools for isolating, manipulating, and reattaching specific anatomical regions (e.g., mandible, maxilla) within 3D models.
  • Implementation of automatic landmark localization on 3D volumetric data.
  • Integration of 3D volume registration for evaluating treatment progress and outcomes.

Main Results:

  • Successful creation of virtual 3D models with automatically identified landmarks.
  • Capability to isolate and manipulate specific craniofacial structures within the 3D model.
  • Enabling of advanced analyses including prospective treatment, end result, and subtraction analysis.

Conclusions:

  • The introduced methods and tools effectively manage 3D orthodontic imaging data.
  • These advancements support a comprehensive approach to orthodontic diagnosis, treatment planning, and outcome evaluation.
  • This technology enhances the potential for precise and data-driven orthodontic care.