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Euclidean space is a mathematical construct that generalizes the properties of two-dimensional and three-dimensional spaces to any number of dimensions, characterized by the notions of distance and angle. It serves as the foundational setting for classical geometry and is defined by a coordinate system where the distance between points is given by the Euclidean distance formula.
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Orthogonal transformations are linear transformations that preserve the dot product, and thus the length of vectors and the angle between them. These transformations are represented by orthogonal matrices, which have the property that their transpose is equal to their inverse.
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An orthogonal region is a geometric area in a multi-dimensional space where all axes are perpendicular to each other, allowing for simplified mathematical operations and analysis. This concept is crucial in fields like linear algebra and computer graphics, where it aids in transformations and optimizations by providing a clear, non-overlapping structure.
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Orthogonal projection is a linear transformation that maps a vector onto a subspace in such a way that the error, or the difference between the vector and its projection, is minimized and orthogonal to the subspace. This concept is fundamental in linear algebra and is widely applied in fields such as computer graphics, signal processing, and statistics for dimensionality reduction and data approximation.
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An affine space is a geometric structure that generalizes the properties of Euclidean spaces, allowing for the definition of points and vectors without a fixed origin. It is characterized by the ability to perform vector addition and scalar multiplication while maintaining the concept of parallelism and affine transformations.
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Linear geometry is a branch of geometry that deals with figures and properties that are defined using linear equations and inequalities, focusing on lines, planes, and their intersections. It serves as the foundation for understanding more complex geometric structures and is essential in fields like computer graphics, physics, and engineering.
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A local coordinate system is a coordinate system that is defined in relation to a specific point or object, often used to simplify calculations or transformations within a localized context. It allows for easier manipulation of objects in a scene by providing a reference frame that moves and rotates with the object itself.
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The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This theorem is fundamental in geometry as it ensures the possibility of forming a triangle and is applicable in various mathematical fields including metric spaces and vector analysis.
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Inner product preservation refers to the property of a transformation, typically a linear map or matrix, that maintains the Inner product (dot product) of vectors after transformation. This property is crucial in various fields such as quantum mechanics and computer graphics, ensuring that angles and lengths are preserved under the transformation, thus maintaining geometric integrity.
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The Heine-Borel theorem states that in Euclidean space, a subset is compact if and only if it is closed and bounded. This theorem is fundamental in real analysis as it characterizes compactness, which is crucial for understanding continuity, convergence, and the behavior of functions on closed intervals.
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The radius of a sphere is the distance from its center to any point on its surface, and it is a fundamental parameter in determining the sphere's geometric properties. It plays a crucial role in calculating the sphere's volume and surface area, as well as in applications across physics and engineering.
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Coordinate systems provide a framework for defining the position of points in space, using a set of numbers called coordinates. They are essential in mathematics, physics, and engineering for describing spatial relationships and transformations between different reference frames.
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A convex body is a compact convex set with non-empty interior in a Euclidean space, meaning it is a shape where, for any two points within the shape, the line segment connecting them is entirely contained within the shape. Convex bodies are fundamental in geometry and optimization, serving as the building blocks for understanding more complex structures and problems in these fields.
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Orthogonal vectors are vectors in a vector space that are perpendicular to each other, meaning their dot product is zero. This property is fundamental in various applications, including simplifying computations in linear algebra and ensuring independence in statistical methods.
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An orthogonal basis of a vector space is a set of vectors that are mutually perpendicular and span the entire space, allowing any vector in the space to be uniquely represented as a linear combination of these basis vectors. This concept simplifies many mathematical computations, such as projections and transformations, due to the orthogonality property that enables easy calculation of coefficients in the linear combination.
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Scalar curvature is a single number at each point on a Riemannian manifold that represents the degree to which the geometry determined by the metric tensor deviates from being flat at that point. It provides a measure of the intrinsic curvature of the manifold, summarizing how volumes of small geodesic balls differ from those in Euclidean space.
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High-dimensional space refers to a mathematical construct where the number of dimensions exceeds three, often used in fields like data science and machine learning to represent complex datasets. As the number of dimensions increases, phenomena such as the 'curse of dimensionality' can arise, making visualization and computation more challenging.
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The orthogonal group, denoted as O(n), is the group of n×n orthogonal matrices, which preserve the Euclidean norm and are characterized by the property that their transpose is equal to their inverse. This group is significant in various fields such as physics and computer science as it describes symmetries and rotations in n-dimensional space while maintaining the structure of geometric objects.
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A probability simplex is a geometric representation of a probability distribution over a finite set of outcomes, where each point within the simplex corresponds to a possible distribution. It is a subset of a Euclidean space defined by the conditions that the coordinates are non-negative and sum to one, reflecting the properties of probabilities.
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A coordinate chart is a mathematical tool used in differential geometry to provide a local coordinate system for a manifold, facilitating the study of its properties by mapping it to Euclidean space. It is essential for defining differentiable structures and performing calculations involving vectors, tensors, and other geometric objects on manifolds.
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A differentiable manifold is a type of manifold that is locally similar to Euclidean space and has a globally defined differential structure, allowing for calculus to be performed on it. This structure enables the application of differential geometry and the study of smooth functions, tangent spaces, and vector fields on the manifold.
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A manifold is a topological space that locally resembles Euclidean space, allowing for the application of calculus and other mathematical tools. Manifolds are fundamental in mathematics and physics, providing the framework for understanding complex structures like curves, surfaces, and higher-dimensional spaces.
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A translation vector is a mathematical tool used to move every point of a shape or object in a specific direction and distance within a coordinate system, without altering its orientation or size. It is fundamental in fields like geometry, physics, and computer graphics for operations involving shifting objects in space.
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Three-dimensional space is a geometric setting in which three values, often referred to as dimensions, are required to determine the position of an element. It is the physical universe we live in, where objects have length, width, and height, allowing for the representation and manipulation of objects in a realistic manner.
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An isometry is a transformation in geometry that preserves distances between points, meaning the original shape and the transformed shape are congruent. Isometries include translations, rotations, reflections, and glide reflections, and are fundamental in understanding symmetry and rigid motions in Euclidean spaces.
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Geometric entities are fundamental elements in geometry that include points, lines, and planes, serving as the building blocks for more complex geometric structures and relationships. Understanding these entities is crucial for exploring spatial concepts, solving geometric problems, and applying mathematical reasoning in various fields such as architecture, engineering, and computer graphics.
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A topological manifold is a topological space that locally resembles Euclidean space, meaning each point has a neighborhood homeomorphic to an open subset of Euclidean space. This property allows for the application of calculus and other analytical methods in a more generalized setting beyond traditional Euclidean spaces.
The three-dimensional coordinate system is a framework for locating points in space using three coordinates, typically labeled as x, y, and z. This system extends the two-dimensional Cartesian coordinate system by adding a third axis, allowing for the representation and analysis of objects and phenomena in three-dimensional space.
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Manifoldness refers to the property of a space that locally resembles Euclidean space, allowing for complex structures to be analyzed in terms of simpler, well-understood geometrical entities. This concept is fundamental in fields such as differential geometry, topology, and theoretical physics, where it aids in understanding the shape and structure of various mathematical and physical phenomena.
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A spherical Coxeter group is a group generated by reflections across hyperplanes in a Euclidean space, where the group acts on a sphere and the corresponding Coxeter diagram is finite. These groups are fundamental in the study of symmetry and geometry, as they classify regular polytopes and tessellations in spherical spaces.
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The area of a parallelogram is calculated by multiplying the base by the height, where the height is the perpendicular distance from the base to the opposite side. This formula highlights the importance of understanding the relationship between the sides and angles of a parallelogram in determining its overall size.
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