Triangle Inequality Theorem
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The triangle inequality theorem is a geometric theorem which states that for any triangle (t), the length of any triangle side must be [less than]] the summation of the other two sides but greater than the mathematical difference between the two triangle sides.
- Context:
- d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z in Metric Space M.
- See: Distance Measure, Metric Space, Euclidean Distance Measure, Great Circle, Inner Product Space, Mathematics, Triangle, Euclidean Geometry, Norm (Mathematics), Law of Cosines.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/triangle_inequality Retrieved:2016-5-1.
- In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. [1] [2] If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that : [math]\displaystyle{ z \leq x + y , }[/math] with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms): : [math]\displaystyle{ \|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| , }[/math] where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in ℝ1, and the triangle inequality expresses a relationship between absolute values. In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either ℝ2 or ℝ3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints. [3]
The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces.
- In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. [1] [2] If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that : [math]\displaystyle{ z \leq x + y , }[/math] with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms): : [math]\displaystyle{ \|\mathbf x + \mathbf y\| \leq \|\mathbf x\| + \|\mathbf y\| , }[/math] where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in ℝ1, and the triangle inequality expresses a relationship between absolute values. In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either ℝ2 or ℝ3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints. [3]
- ↑ Wolfram MathWorld - http://mathworld.wolfram.com/TriangleInequality.html
- ↑ |isbn=0-471-41825-0 |year=2001 |publisher=Wiley-IEEE}}
- ↑
1998
- (Fagin & Stockmeyer, 1998) ⇒ Ronald Fagin, and Larry Stockmeyer. (1998). “Relaxing the Triangle Inequality in Pattern Matching.” In: International Journal of Computer Vision, 30(3). doi:10.1023/A:1008023416823
- QUOTE: Any notion of “closeness” in pattern matching should have the property that if A is close to B, and B is close to C, then A is close to C. Traditionally, this property is attained because of the triangle inequality (d(A, C) ≤ d(A, B) + d(B, C), where d represents a notion of distance). However, the full power of the triangle inequality is not needed for this property to hold. Instead, a “relaxed triangle inequality” suffices, of the form d(A, C) ≤ c(d(A, B) + d(B, C)), where c is a constant that is not too large. In this paper, we show that one of the measures used for distances between shapes in (an experimental version of) IBM's QBIC1 ("Query by Image Content") system (Niblack et al., 1993) satisfies a relaxed triangle inequality, although it does not satisfy the triangle inequality.