104-436: In geometry , a cardioid (from Greek καρδιά (kardiá) 'heart') is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp . It is also a type of sinusoidal spiral , and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as
208-644: A . {\displaystyle {\begin{aligned}|PQ|&=r(\varphi )+r(\varphi +\pi )\\&=2a(1-\cos \varphi )+2a(1-\cos(\varphi +\pi ))=\cdots =4a\end{aligned}}.} For the proof the representation in the complex plane (see above) is used. For the points P : p ( φ ) = a ( − e i 2 φ + 2 e i φ − 1 ) {\displaystyle P:\ p(\varphi )=a\,\left(-e^{i2\varphi }+2e^{i\varphi }-1\right)} and Q : p ( φ + π ) =
312-493: A ( − e i 2 ( φ + π ) + 2 e i ( φ + π ) − 1 ) = a ( − e i 2 φ − 2 e i φ − 1 ) , {\displaystyle Q:\ p(\varphi +\pi )=a\,\left(-e^{i2(\varphi +\pi )}+2e^{i(\varphi +\pi )}-1\right)=a\,\left(-e^{i2\varphi }-2e^{i\varphi }-1\right),}
416-405: A {\displaystyle -a} by the same angle φ {\displaystyle \varphi } : p ( φ ) = Φ − ( Φ + ( 0 ) ) = Φ − ( a − a e i φ ) = − a + ( a −
520-662: A ∫ 0 π 1 2 ( 1 − cos φ ) d φ = 8 a ∫ 0 π sin ( φ 2 ) d φ = 16 a . {\displaystyle L=2\int _{0}^{\pi }{\sqrt {r(\varphi )^{2}+(r'(\varphi ))^{2}}}\;d\varphi =\cdots =8a\int _{0}^{\pi }{\sqrt {{\tfrac {1}{2}}(1-\cos \varphi )}}\;d\varphi =8a\int _{0}^{\pi }\sin \left({\tfrac {\varphi }{2}}\right)d\varphi =16a.} The radius of curvature ρ {\displaystyle \rho } of
624-1498: A ( − sin ( 2 φ ) + 2 sin φ ) = 2 a ( 1 − cos φ ) ⋅ sin φ . {\displaystyle {\begin{array}{cclcccc}x(\varphi )&=&a\;(-\cos(2\varphi )+2\cos \varphi -1)&=&2a(1-\cos \varphi )\cdot \cos \varphi &&\\y(\varphi )&=&a\;(-\sin(2\varphi )+2\sin \varphi )&=&2a(1-\cos \varphi )\cdot \sin \varphi &.&\end{array}}} (The trigonometric identities e i φ = cos φ + i sin φ , ( cos φ ) 2 + ( sin φ ) 2 = 1 , {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi ,\ (\cos \varphi )^{2}+(\sin \varphi )^{2}=1,} cos ( 2 φ ) = ( cos φ ) 2 − ( sin φ ) 2 , {\displaystyle \cos(2\varphi )=(\cos \varphi )^{2}-(\sin \varphi )^{2},} and sin ( 2 φ ) = 2 sin φ cos φ {\displaystyle \sin(2\varphi )=2\sin \varphi \cos \varphi } were used.) For
728-690: A 2 sin 2 φ 2 ] 3 2 24 a 2 sin 2 φ 2 = 8 3 a sin φ 2 . {\displaystyle \rho (\varphi )=\cdots ={\frac {\left[16a^{2}\sin ^{2}{\frac {\varphi }{2}}\right]^{\frac {3}{2}}}{24a^{2}\sin ^{2}{\frac {\varphi }{2}}}}={\frac {8}{3}}a\sin {\frac {\varphi }{2}}\ .} The points P : p ( φ ) , Q : p ( φ + π ) {\displaystyle P:p(\varphi ),\;Q:p(\varphi +\pi )} are on
832-696: A 2 ( 1 − cos φ ) 2 d φ = ⋯ = 4 a 2 ⋅ 3 2 π = 6 π a 2 . {\displaystyle A=2\cdot {\tfrac {1}{2}}\int _{0}^{\pi }{(r(\varphi ))^{2}}\;d\varphi =\int _{0}^{\pi }{4a^{2}(1-\cos \varphi )^{2}}\;d\varphi =\cdots =4a^{2}\cdot {\tfrac {3}{2}}\pi =6\pi a^{2}.} L = 2 ∫ 0 π r ( φ ) 2 + ( r ′ ( φ ) ) 2 d φ = ⋯ = 8
936-448: A e i φ + a ) e i φ = a ( − e i 2 φ + 2 e i φ − 1 ) . {\displaystyle p(\varphi )=\Phi _{-}(\Phi _{+}(0))=\Phi _{-}\left(a-ae^{i\varphi }\right)=-a+\left(a-ae^{i\varphi }+a\right)e^{i\varphi }=a\;\left(-e^{i2\varphi }+2e^{i\varphi }-1\right).} From here one gets
1040-665: A sin 2 φ 2 sin φ + 8 3 a sin φ 2 ⋅ cos 3 2 φ = ⋯ = 4 3 a cos 2 φ 2 sin φ . {\displaystyle Y(\varphi )=4a\sin ^{2}{\tfrac {\varphi }{2}}\sin \varphi +{\tfrac {8}{3}}a\sin {\tfrac {\varphi }{2}}\cdot \cos {\tfrac {3}{2}}\varphi =\cdots ={\tfrac {4}{3}}a\cos ^{2}{\tfrac {\varphi }{2}}\sin \varphi \,.} These equations describe
1144-668: A ( 1 − cos φ ) sin φ = 4 a sin 2 φ 2 sin φ {\displaystyle y(\varphi )=2a(1-\cos \varphi )\sin \varphi =4a\sin ^{2}{\tfrac {\varphi }{2}}\sin \varphi } the unit normal is n → ( φ ) = ( − sin 3 2 φ , cos 3 2 φ ) {\displaystyle {\vec {n}}(\varphi )=(-\sin {\tfrac {3}{2}}\varphi ,\cos {\tfrac {3}{2}}\varphi )} and
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#17327878127711248-452: A + 2 a cos φ , 2 a sin φ ) {\displaystyle (2a+2a\cos \varphi ,2a\sin \varphi )} has the equation ( x − 2 a ) ⋅ cos φ + y ⋅ sin φ = 2 a . {\displaystyle (x-2a)\cdot \cos \varphi +y\cdot \sin \varphi =2a\,.} The foot of
1352-614: A sin φ 2 ⋅ sin 3 2 φ = ⋯ = 4 3 a cos 2 φ 2 cos φ − 4 3 a , {\displaystyle X(\varphi )=4a\sin ^{2}{\tfrac {\varphi }{2}}\cos \varphi -{\tfrac {8}{3}}a\sin {\tfrac {\varphi }{2}}\cdot \sin {\tfrac {3}{2}}\varphi =\cdots ={\tfrac {4}{3}}a\cos ^{2}{\tfrac {\varphi }{2}}\cos \varphi -{\tfrac {4}{3}}a\,,} Y ( φ ) = 4
1456-401: A chord through the cusp (=origin). Hence | P Q | = r ( φ ) + r ( φ + π ) = 2 a ( 1 − cos φ ) + 2 a ( 1 − cos ( φ + π ) ) = ⋯ = 4
1560-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as
1664-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to
1768-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes
1872-496: A cardioid (any 2d plane containing the 3d straight line of the microphone body). In three dimensions, the cardioid is shaped like an apple centred around the microphone which is the "stalk" of the apple. Let a {\displaystyle a} be the common radius of the two generating circles with midpoints ( − a , 0 ) , ( a , 0 ) {\displaystyle (-a,0),(a,0)} , φ {\displaystyle \varphi }
1976-1106: A cardioid a third as large, rotated 180 degrees and shifted along the x-axis by − 4 3 a {\displaystyle -{\tfrac {4}{3}}a} . (Trigonometric formulae were used: sin 3 2 φ = sin φ 2 cos φ + cos φ 2 sin φ , cos 3 2 φ = ⋯ , sin φ = 2 sin φ 2 cos φ 2 , cos φ = ⋯ . {\displaystyle \sin {\tfrac {3}{2}}\varphi =\sin {\tfrac {\varphi }{2}}\cos \varphi +\cos {\tfrac {\varphi }{2}}\sin \varphi \ ,\ \cos {\tfrac {3}{2}}\varphi =\cdots ,\ \sin \varphi =2\sin {\tfrac {\varphi }{2}}\cos {\tfrac {\varphi }{2}},\ \cos \varphi =\cdots \ .} ) An orthogonal trajectory of
2080-473: A circle and O {\displaystyle O} a point on the perimeter of this circle. The following is true: Hence a cardioid is a special pedal curve of a circle. In a Cartesian coordinate system circle k {\displaystyle k} may have midpoint ( 2 a , 0 ) {\displaystyle (2a,0)} and radius 2 a {\displaystyle 2a} . The tangent at circle point ( 2
2184-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,
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#17327878127712288-798: A curve in polar coordinates with equation r = r ( φ ) {\displaystyle r=r(\varphi )} is (s. curvature ) ρ ( φ ) = [ r ( φ ) 2 + r ˙ ( φ ) 2 ] 3 / 2 r ( φ ) 2 + 2 r ˙ ( φ ) 2 − r ( φ ) r ¨ ( φ ) . {\displaystyle \rho (\varphi )={\frac {\left[r(\varphi )^{2}+{\dot {r}}(\varphi )^{2}\right]^{3/2}}{r(\varphi )^{2}+2{\dot {r}}(\varphi )^{2}-r(\varphi ){\ddot {r}}(\varphi )}}\ .} For
2392-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In
2496-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,
2600-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry
2704-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in
2808-814: A pencil of curves is a curve which intersects any curve of the pencil orthogonally. For cardioids the following is true: (The second pencil can be considered as reflections at the y-axis of the first one. See diagram.) For a curve given in polar coordinates by a function r ( φ ) {\displaystyle r(\varphi )} the following connection to Cartesian coordinates hold: x ( φ ) = r ( φ ) cos φ , y ( φ ) = r ( φ ) sin φ {\displaystyle {\begin{aligned}x(\varphi )&=r(\varphi )\cos \varphi \,,\\y(\varphi )&=r(\varphi )\sin \varphi \end{aligned}}} and for
2912-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received
3016-528: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to
3120-417: A point z {\displaystyle z} (complex number) by e i φ {\displaystyle e^{i\varphi }} . Hence A point p ( φ ) {\displaystyle p(\varphi )} of the cardioid is generated by rotating the origin around point a {\displaystyle a} and subsequently rotating around −
3224-602: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies
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3328-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating
3432-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c. 1900 , with
3536-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in
3640-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of
3744-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing
3848-540: Is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model
3952-468: Is a parabola (s. parabola in polar coordinates ) with the equation x = 1 2 ( y 2 − 1 ) {\textstyle x={\tfrac {1}{2}}\left(y^{2}-1\right)} in Cartesian coordinates. Remark: Not every inverse curve of a parabola is a cardioid. For example, if a parabola is inverted across a circle whose center lies at the vertex of
4056-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in
4160-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood
4264-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are
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4368-617: Is equivalent to F ( x , y , t ) = x 2 + y 2 + 2 x ( 1 − cos t ) − 2 y sin t = 0 . {\displaystyle F(x,y,t)=x^{2}+y^{2}+2x\;(1-\cos t)-2y\;\sin t=0\;.} The second envelope condition is F t ( x , y , t ) = 2 x sin t − 2 y cos t = 0. {\displaystyle F_{t}(x,y,t)=2x\;\sin t-2y\;\cos t=0.} One easily checks that
4472-799: Is identical to the angle parameter of the cardioid. A similar and simple method to draw a cardioid uses a pencil of lines . It is due to L. Cremona : The following consideration uses trigonometric formulae for cos α + cos β {\displaystyle \cos \alpha +\cos \beta } , sin α + sin β {\displaystyle \sin \alpha +\sin \beta } , 1 + cos 2 α {\displaystyle 1+\cos 2\alpha } , cos 2 α {\displaystyle \cos 2\alpha } , and sin 2 α {\displaystyle \sin 2\alpha } . In order to keep
4576-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given
4680-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study
4784-500: Is part of the line with equation (see previous section) cos ( 3 2 φ ) x + sin ( 3 2 φ ) y = 4 ( cos 1 2 φ ) 3 , {\displaystyle \cos \left({\tfrac {3}{2}}\varphi \right)x+\sin \left({\tfrac {3}{2}}\varphi \right)y=4\left(\cos {\tfrac {1}{2}}\varphi \right)^{3}\,,} which
4888-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,
4992-434: Is tangent of the cardioid with polar equation r = 2 ( 1 + cos φ ) {\displaystyle r=2(1+\cos \varphi )} from the previous section. Remark: For such considerations usually multiple reflections at the circle are neglected. The Cremona generation of a cardioid should not be confused with the following generation: Let be k {\displaystyle k}
5096-1237: Is the envelope condition . Note that F t {\displaystyle F_{t}} means the partial derivative for parameter t {\displaystyle t} . Let c {\displaystyle c} be the circle with midpoint ( − 1 , 0 ) {\displaystyle (-1,0)} and radius 1 {\displaystyle 1} . Then c {\displaystyle c} has parametric representation ( − 1 + cos t , sin t ) {\displaystyle (-1+\cos t,\sin t)} . The pencil of circles with centers on c {\displaystyle c} containing point O = ( 0 , 0 ) {\displaystyle O=(0,0)} can be represented implicitly by F ( x , y , t ) = ( x + 1 − cos t ) 2 + ( y − sin t ) 2 − ( 2 − 2 cos t ) = 0 , {\displaystyle F(x,y,t)=(x+1-\cos t)^{2}+(y-\sin t)^{2}-(2-2\cos t)=0,} which
5200-663: Is the pencil of secant lines of a circle (s. above) and F t ( x , y , t ) = − 3 2 sin ( 3 2 t ) x + 3 2 cos ( 3 2 t ) y + 3 cos ( 1 2 t ) sin t = 0 . {\displaystyle F_{t}(x,y,t)=-{\tfrac {3}{2}}\sin \left({\tfrac {3}{2}}t\right)x+{\tfrac {3}{2}}\cos \left({\tfrac {3}{2}}t\right)y+3\cos \left({\tfrac {1}{2}}t\right)\sin t=0\,.} For fixed parameter t both
5304-563: Is the polar equation of a cardioid. Remark: If point O {\displaystyle O} is not on the perimeter of the circle k {\displaystyle k} , one gets a limaçon of Pascal . The evolute of a curve is the locus of centers of curvature. In detail: For a curve x → ( s ) = c → ( s ) {\displaystyle {\vec {x}}(s)={\vec {c}}(s)} with radius of curvature ρ ( s ) {\displaystyle \rho (s)}
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#17327878127715408-753: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs
5512-690: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c. 1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated
5616-523: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c. 1230 – c. 1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first
5720-518: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with
5824-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of
5928-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,
6032-631: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing
6136-496: The 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into
6240-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by
6344-590: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in
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#17327878127716448-796: The calculations simple, the proof is given for the cardioid with polar representation r = 2 ( 1 + cos φ ) {\displaystyle r=2(1\mathbin {\color {red}+} \cos \varphi )} ( § Cardioids in different positions ). From the parametric representation x ( φ ) = 2 ( 1 + cos φ ) cos φ , y ( φ ) = 2 ( 1 + cos φ ) sin φ {\displaystyle {\begin{aligned}x(\varphi )&=2(1+\cos \varphi )\cos \varphi ,\\y(\varphi )&=2(1+\cos \varphi )\sin \varphi \end{aligned}}} one gets
6552-422: The cardioid r ( φ ) = 2 a ( 1 − cos φ ) = 4 a sin 2 ( φ 2 ) {\displaystyle r(\varphi )=2a(1-\cos \varphi )=4a\sin ^{2}\left({\tfrac {\varphi }{2}}\right)} one gets ρ ( φ ) = ⋯ = [ 16
6656-486: The cardioid as defined above the following formulas hold: The proofs of these statements use in both cases the polar representation of the cardioid. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area) A = 2 ⋅ 1 2 ∫ 0 π ( r ( φ ) ) 2 d φ = ∫ 0 π 4
6760-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry
6864-513: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c. 287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of
6968-926: The derivatives d x d φ = r ′ ( φ ) cos φ − r ( φ ) sin φ , d y d φ = r ′ ( φ ) sin φ + r ( φ ) cos φ . {\displaystyle {\begin{aligned}{\frac {dx}{d\varphi }}&=r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi \,,\\{\frac {dy}{d\varphi }}&=r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi \,.\end{aligned}}} Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría ) 'land measurement'; from γῆ ( gê ) 'earth, land' and μέτρον ( métron ) 'a measure')
7072-948: The equation of the secant line passing the two points ( 1 + 3 cos θ , 3 sin θ ) , ( 1 + 3 cos 2 θ , 3 sin 2 θ ) ) {\displaystyle (1+3\cos \theta ,3\sin \theta ),\ (1+3\cos {\color {red}2}\theta ,3\sin {\color {red}2}\theta ))} one gets: ( sin θ − sin 2 θ ) x + ( cos 2 θ − sin θ ) y = − 2 cos θ − sin ( 2 θ ) . {\displaystyle (\sin \theta -\sin 2\theta )x+(\cos 2\theta -\sin \theta )y=-2\cos \theta -\sin(2\theta )\,.} With help of trigonometric formulae and
7176-662: The equation of the tangent can be rewritten as: cos ( 3 2 φ ) ⋅ x + sin ( 3 2 φ ) ⋅ y = 4 ( cos 1 2 φ ) 3 0 < φ < 2 π , φ ≠ π . {\displaystyle \cos({\tfrac {3}{2}}\varphi )\cdot x+\sin \left({\tfrac {3}{2}}\varphi \right)\cdot y=4\left(\cos {\tfrac {1}{2}}\varphi \right)^{3}\quad 0<\varphi <2\pi ,\ \varphi \neq \pi .} For
7280-564: The equations represent lines. Their intersection point is x ( t ) = 2 ( 1 + cos t ) cos t , y ( t ) = 2 ( 1 + cos t ) sin t , {\displaystyle x(t)=2(1+\cos t)\cos t,\quad y(t)=2(1+\cos t)\sin t,} which is a point of the cardioid with polar equation r = 2 ( 1 + cos t ) . {\displaystyle r=2(1+\cos t).} The considerations made in
7384-396: The evolute has the representation X → ( s ) = c → ( s ) + ρ ( s ) n → ( s ) . {\displaystyle {\vec {X}}(s)={\vec {c}}(s)+\rho (s){\vec {n}}(s).} with n → ( s ) {\displaystyle {\vec {n}}(s)}
7488-428: The field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits
7592-520: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c. 355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as
7696-607: The following simple method to draw a cardioid: The envelope of the pencil of implicitly given curves F ( x , y , t ) = 0 {\displaystyle F(x,y,t)=0} with parameter t {\displaystyle t} consists of such points ( x , y ) {\displaystyle (x,y)} which are solutions of the non-linear system F ( x , y , t ) = 0 , F t ( x , y , t ) = 0 , {\displaystyle F(x,y,t)=0,\quad F_{t}(x,y,t)=0,} which
7800-526: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between
7904-598: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning
8008-537: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including
8112-552: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived
8216-436: The midpoint of the chord P Q {\displaystyle PQ} is M : 1 2 ( p ( φ ) + p ( φ + π ) ) = ⋯ = − a − a e i 2 φ {\displaystyle M:\ {\tfrac {1}{2}}(p(\varphi )+p(\varphi +\pi ))=\cdots =-a-ae^{i2\varphi }} which lies on
8320-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of
8424-429: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,
8528-1306: The normal vector n → = ( y ˙ , − x ˙ ) T {\displaystyle {\vec {n}}=\left({\dot {y}},-{\dot {x}}\right)^{\mathsf {T}}} . The equation of the tangent y ˙ ( φ ) ⋅ ( x − x ( φ ) ) − x ˙ ( φ ) ⋅ ( y − y ( φ ) ) = 0 {\displaystyle {\dot {y}}(\varphi )\cdot (x-x(\varphi ))-{\dot {x}}(\varphi )\cdot (y-y(\varphi ))=0} is: ( cos 2 φ + cos φ ) ⋅ x + ( sin 2 φ + sin φ ) ⋅ y = 2 ( 1 + cos φ ) 2 . {\displaystyle (\cos 2\varphi +\cos \varphi )\cdot x+(\sin 2\varphi +\sin \varphi )\cdot y=2(1+\cos \varphi )^{2}\,.} With help of trigonometric formulae and subsequent division by cos 1 2 φ {\textstyle \cos {\frac {1}{2}}\varphi } ,
8632-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of
8736-428: The parabola, then the result is a cissoid of Diocles . In the previous section if one inverts additionally the tangents of the parabola one gets a pencil of circles through the center of inversion (origin). A detailed consideration shows: The midpoints of the circles lie on the perimeter of the fixed generator circle. (The generator circle is the inverse curve of the parabola's directrix.) This property gives rise to
8840-410: The parametric representation above: x ( φ ) = a ( − cos ( 2 φ ) + 2 cos φ − 1 ) = 2 a ( 1 − cos φ ) ⋅ cos φ y ( φ ) =
8944-738: The perimeter of the circle with midpoint − a {\displaystyle -a} and radius a {\displaystyle a} (see picture). For the example shown in the graph the generator circles have radius a = 1 2 {\textstyle a={\frac {1}{2}}} . Hence the cardioid has the polar representation r ( φ ) = 1 − cos φ {\displaystyle r(\varphi )=1-\cos \varphi } and its inverse curve r ( φ ) = 1 1 − cos φ , {\displaystyle r(\varphi )={\frac {1}{1-\cos \varphi }},} which
9048-409: The perpendicular from point O {\displaystyle O} on the tangent is point ( r cos φ , r sin φ ) {\displaystyle (r\cos \varphi ,r\sin \varphi )} with the still unknown distance r {\displaystyle r} to the origin O {\displaystyle O} . Inserting
9152-514: The physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem , a problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During
9256-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of
9360-458: The point into the equation of the tangent yields ( r cos φ − 2 a ) cos φ + r sin 2 φ = 2 a → r = 2 a ( 1 + cos φ ) {\displaystyle (r\cos \varphi -2a)\cos \varphi +r\sin ^{2}\varphi =2a\quad \rightarrow \quad r=2a(1+\cos \varphi )} which
9464-463: The points of the cardioid with the parametric representation x ( t ) = 2 ( 1 − cos t ) cos t , y ( t ) = 2 ( 1 − cos t ) sin t {\displaystyle x(t)=2(1-\cos t)\cos t,\quad y(t)=2(1-\cos t)\sin t} fulfill the non-linear system above. The parameter t {\displaystyle t}
9568-983: The previous section give a proof that the caustic of a circle with light source on the perimeter of the circle is a cardioid. As in the previous section the circle may have midpoint ( 1 , 0 ) {\displaystyle (1,0)} and radius 3 {\displaystyle 3} . Its parametric representation is c ( φ ) = ( 1 + 3 cos φ , 3 sin φ ) . {\displaystyle c(\varphi )=(1+3\cos \varphi ,3\sin \varphi )\ .} The tangent at circle point C : k ( φ ) {\displaystyle C:\ k(\varphi )} has normal vector n → t = ( cos φ , sin φ ) T {\displaystyle {\vec {n}}_{t}=(\cos \varphi ,\sin \varphi )^{\mathsf {T}}} . Hence
9672-482: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that
9776-474: The radius of curvature ρ ( φ ) = 8 3 a sin φ 2 . {\displaystyle \rho (\varphi )={\tfrac {8}{3}}a\sin {\tfrac {\varphi }{2}}\,.} Hence the parametric equations of the evolute are X ( φ ) = 4 a sin 2 φ 2 cos φ − 8 3
9880-668: The reflected ray has the normal vector n → r = ( cos 3 2 φ , sin 3 2 φ ) T {\displaystyle {\vec {n}}_{r}=\left(\cos {\color {red}{\tfrac {3}{2}}}\varphi ,\sin {\color {red}{\tfrac {3}{2}}}\varphi \right)^{\mathsf {T}}} (see graph) and contains point C : ( 1 + 3 cos φ , 3 sin φ ) {\displaystyle C:\ (1+3\cos \varphi ,3\sin \varphi )} . The reflected ray
9984-482: The rolling angle and the origin the starting point (see picture). One gets the A proof can be established using complex numbers and their common description as the complex plane . The rolling movement of the black circle on the blue one can be split into two rotations. In the complex plane a rotation around point 0 {\displaystyle 0} (the origin) by an angle φ {\displaystyle \varphi } can be performed by multiplying
10088-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid
10192-464: The set of points of reflections of a fixed point on a circle through all tangents to the circle. The name was coined by Giovanni Salvemini in 1741 but the cardioid had been the subject of study decades beforehand. Although named for its heart-like form, it is shaped more like the outline of the cross-section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles
10296-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry
10400-753: The subsequent division by sin 1 2 θ {\textstyle \sin {\frac {1}{2}}\theta } the equation of the secant line can be rewritten by: cos ( 3 2 θ ) ⋅ x + sin ( 3 2 θ ) ⋅ y = 4 ( cos 1 2 θ ) 3 0 < θ < 2 π . {\displaystyle \cos \left({\tfrac {3}{2}}\theta \right)\cdot x+\sin \left({\tfrac {3}{2}}\theta \right)\cdot y=4\left(\cos {\tfrac {1}{2}}\theta \right)^{3}\quad 0<\theta <2\pi .} Despite
10504-544: The suitably oriented unit normal. For a cardioid one gets: For the cardioid with parametric representation x ( φ ) = 2 a ( 1 − cos φ ) cos φ = 4 a sin 2 φ 2 cos φ , {\displaystyle x(\varphi )=2a(1-\cos \varphi )\cos \varphi =4a\sin ^{2}{\tfrac {\varphi }{2}}\cos \varphi \,,} y ( φ ) = 2
10608-409: The theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and
10712-925: The two angles φ , θ {\displaystyle \varphi ,\theta } have different meanings (s. picture) one gets for φ = θ {\displaystyle \varphi =\theta } the same line. Hence any secant line of the circle, defined above, is a tangent of the cardioid, too: Remark: The proof can be performed with help of the envelope conditions (see previous section) of an implicit pencil of curves: F ( x , y , t ) = cos ( 3 2 t ) x + sin ( 3 2 t ) y − 4 ( cos 1 2 t ) 3 = 0 {\displaystyle F(x,y,t)=\cos \left({\tfrac {3}{2}}t\right)x+\sin \left({\tfrac {3}{2}}t\right)y-4\left(\cos {\tfrac {1}{2}}t\right)^{3}=0}
10816-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in
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