# إحداثيات متجانسة

منحنى بيزييه كسري – منحنى متعدد الأطراف مُعرَّف بإحداثيات متجانسة (أزرق) وإسقاطه على مستوى – منحنى كسري (أحمر)

في الرياضيات، الإحداثيات المتجانسة Homogeneous coordinates التي تم تقديم مفهومها من قبل أوگوست فرديناند موبيوس في عام 1827 تسمح بتمثيل التحويلات الأفينية بشكل بسيط باستخدام المصفوفات. كما تسهل إجراء الحسابات في فضاء الإسقاط كما يسهل نظام الإحداثيات الديكارتية هذه الحسابات في الفضاء الإقليدي.

تكتب الإحداثيات المتجانسة لنقطة تنتمي إلى فضاء الإسقاط ذو البعد n بالصيغة (x : y : z : ... : w) على شكل متجه من صف واحد طوله n+1.

For example, the Cartesian point (1, 2) can be represented in homogeneous coordinates as (1, 2, 1) or (2, 4, 2). The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.

The equation of a line through the origin (0, 0) may be written nx + my = 0 where n and m are not both 0. In parametric form this can be written x = mt, y = −nt. Let Z = 1/t, so the coordinates of a point on the line may be written (m/Z, −n/Z). In homogeneous coordinates this becomes (m, −n, Z). In the limit, as t approaches infinity, in other words, as the point moves away from the origin, Z approaches 0 and the homogeneous coordinates of the point become (m, −n, 0). Thus we define (m, −n, 0) as the homogeneous coordinates of the point at infinity corresponding to the direction of the line nx + my = 0. As any line of the Euclidean plane is parallel to a line passing through the origin, and since parallel lines have the same point at infinity, the infinite point on every line of the Euclidean plane has been given homogeneous coordinates.

للتلخيص:

• أي نقطة في مستوى الإسقاط تُمثـَّل بالثلاثي (X, Y, Z)، تسمى إحداثيات متجانسة أو إحداثيات إسقاطية للنقطة، حيث X, Y and Z are not all 0.
• The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.
• Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying all the coordinates by the same non-zero constant.
• When Z is not 0 the point represented is the point (X/Z, Y/Z) in the Euclidean plane.
• When Z is 0 the point represented is a point at infinity.

Note that the triple (0, 0, 0) is omitted and does not represent any point. The origin is represented by (0, 0, 1).[1]

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## التجانس

Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say f(x, y, z), does not determine a function defined on points as with Cartesian coordinates. But a condition f(x, y, z) = 0 defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is homogeneous. Specifically, suppose there is a k such that

${\displaystyle f(\lambda x,\lambda y,\lambda z)=\lambda ^{k}f(x,y,z).\,}$

If a set of coordinates represent the same point as (x, y, z) then it can be written x, λy, λz) for some non-zero value of λ. Then

${\displaystyle f(x,y,z)=0\iff f(\lambda x,\lambda y,\lambda z)=\lambda ^{k}f(x,y,z)=0.}$

A polynomial g(x, y) of degree k can be turned into a homogeneous polynomial by replacing x with x/z, y with y/z and multiplying by zk, in other words by defining

${\displaystyle f(x,y,z)=z^{k}g(x/z,y/z).\,}$

The resulting function f is a polynomial so it makes sense to extend its domain to triples where z = 0. The process can be reversed by setting z = 1, or

${\displaystyle g(x,y)=f(x,y,1).\,}$

The equation f(x, y, z) = 0 can then be thought of as the homogeneous form of g(x, y) = 0 and it defines the same curve when restricted to the Euclidean plane. For example, the homogeneous form of the equation of the line ax + by + c = 0 is ax + by + cz = 0.[2]

## إحداثيات پلوكر

Assigning coordinates to lines in projective 3-space is more complicated since it would seem that at total of 8 coordinates, either the coordinates of two points which lie on the line or two planes whose intersection is the line. A useful method, due to Julius Plücker, creates a set of six coordinate as the determinants xiyjxjyi (1 ≤ i < j ≤ 4) from the homogeneous coordinates of two points (x1, x2, x3, x4) and (y1, y2, y3, y4) on the line. The Plücker embedding is the generalization of this to create homogeneous coordinates of elements of any dimension m in a projective space of dimension n.[3][4]

## التطبيق في مبرهنة بيزو

تتوقع مبرهنة بيزو أن عدد نقاط التقاطع لمنحنيين يساوي حاصل ضرب درجتيهما (بافتراض an algebraically closed field and with certain conventions followed for counting intersection multiplicities). Bézout's theorem predicts there is one point of intersection of two lines and in general this is true, but when the lines are parallel the point of intersection is infinite. Homogeneous coordinates are used to locate the point of intersection in this case. Similarly, Bézout's theorem predicts that a line will intersect a conic at two points, but in some cases one or both of the points is infinite and homogeneous coordinates must be used to locate them. For example, y = x2 and x = 0 have only one point of intersection in the finite (affine) plane. To find the other point of intersection, convert the equations into homogeneous form, yz = x2 and x = 0. This produces x = yz = 0 and, assuming not all of x, y and z are 0, the solutions are x = y = 0, z ≠ 0 and x = z = 0, y ≠ 0. This first solution is the point (0, 0) in Cartesian coordinates, the finite point of intersection. The second solution gives the homogeneous coordinates (0, 1, 0) which corresponds to the direction of the y-axis. For the equations xy = 1 and x = 0 there are no finite points of intersection. Converting the equations into homogeneous form gives xy = z2 and x = 0. Solving produces the equation z2 = 0 which has a double root at z = 0. From the original equation, x = 0, so y ≠ 0 since at least one coordinate must be non-zero. Therefore, (0, 1, 0) is the point of intersection counted with multiplicity 2 in agreement with the theorem.[5]

## النقاط الدائرية

The homogeneous form for the equation of a circle in the real or complex projective plane is x2 + y2 + 2axz + 2byz + cz2 = 0. The intersection of this curve with the line at infinity can be found by setting z = 0. This produces the equation x2 + y2 = 0 which has two solutions over the complex numbers, giving rise to the points with homogeneous coordinates (1, i, 0) and (1, −i, 0) in the complex projective plane. These points are called the circular points at infinity and can be regarded as the common points of intersection of all circles. This can be generalized to curves of higher order as circular algebraic curves.[6]

## تغيير نظام الإحداثيات

Just as the selection of axes in the Cartesian coordinate system is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore, it is useful to know how the different systems are related to each other.

Let (x, y, z) be the homogeneous coordinates of a point in the projective plane and for a fixed matrix

${\displaystyle A={\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}},}$

with det(A) ≠ 0, define a new set of coordinates (X, Y, Z) by the equation

${\displaystyle {\begin{pmatrix}X\\Y\\Z\end{pmatrix}}=A{\begin{pmatrix}x\\y\\z\end{pmatrix}}.}$

Multiplication of (x, y, z) by a scalar results in the multiplication of (X, Y, Z) by the same scalar, and X, Y and Z cannot be all 0 unless x, y and z are all zero since A is nonsingular. So (X, Y, Z) are a new system of homogeneous coordinates for points in the projective plane. If z is fixed at 1 then

${\displaystyle X=ax+by+c,\,Y=dx+ey+f,\,Z=gx+hy+i}$

are proportional to the signed distances from the point to the lines

${\displaystyle ax+by+c=0,\,dx+ey+f=0,\,gx+hy+i=0.}$

(The signed distance is the distance multiplied by a sign 1 or −1 depending on which side of the line the point lies.) Note that for a = b = 0 the value of X is simply a constant, and similarly for Y and Z.

الخطوط الثلاث،

${\displaystyle ax+by+cz=0,\,dx+ey+fz=0,\,gx+hy+iz=0}$

in homogeneous coordinates, or

${\displaystyle X=0,\,Y=0,\,Z=0}$

in the (X, Y, Z) system, form a triangle called the triangle of reference for the system.[7]

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## الإحداثيات الثقلية

الصيغة الأصلية من موبيوس للإحداثيات المتجانسة حددت موقع نقطة ما بأنه مركز الثقل (أو barycenter) لنظام من ثلاث كتل نقطية على رؤوس مثلث ثابت. النقاط داخل المثلث تُمثـَّل بكتل موجبة والنقاط خارج المثلث تُمثـَّل بالسماح لكتل سالبة. ضرب الكتل في النظام بمقياس لا يؤثر على مركز الثقل، ولذلك فهذه حالة خاصة من نظام إحداثيات متجانسة.

## الإحداثيات ثلاثية الخطوط

Let l, m, n be three lines in the plane and define a set of coordinates X, Y and Z of a point p as the signed distances from p to these three lines. These are called the trilinear coordinates of p with respect to the triangle whose vertices are the pairwise intersections of the lines. Strictly speaking these are not homogeneous, since the values of X, Y and Z are determined exactly, not just up to proportionality. There is a linear relationship between them however, so these coordinates can be made homogeneous by allowing multiples of (X, Y, Z) to represent the same point. More generally, X, Y and Z can be defined as constants p, r and q times the distances to l, m and n, resulting in a different system of homogeneous coordinates with the same triangle of reference. This is, in fact, the most general type of system of homogeneous coordinates for points in the plane if none of the lines is the line at infinity.[8]

## الاستخدام في الرسم الحاسوبي

Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as translation, rotation, scaling and perspective projection to be represented as a matrix by which the vector is multiplied. By the chain rule, any sequence of such operations can be multiplied out into a single matrix, allowing simple and efficient processing. By contrast, using Cartesian coordinates, translations and perspective projection cannot be expressed as matrix multiplications, though other operations can. Modern OpenGL and Direct3D graphics cards take advantage of homogeneous coordinates to implement a vertex shader efficiently using vector processors with 4-element registers.[9][10]

## الهامش

1. ^ For the section: Jones 1912
2. ^ For the section: Miranda 1995, p. 14 and Jones 1912, p. 120
3. ^ Wilczynski 1906, p. 50
4. ^ Bôcher 1907, p. 110
5. ^ Jones 1912, pp. 117–118, 122 with simplified examples.
6. ^ Jones 1912, p. 204
7. ^ Briot & Bouquet 1896
8. ^ Jones 1912, pp. 452 ff
9. ^ http://msdn.microsoft.com/en-us/library/bb206341(VS.85).aspx
10. ^ Shreiner, Dave; Woo, Mason; Neider, Jackie; Davis, Tom; "OpenGL Programming Guide", 4th Edition, ISBN 978-0-321-17348-5, published December 2004. Page 38 and Appendix F (pp. 697-702) Discuss how OpenGL uses homogeneous coordinates in its rendering pipeline. Page 2 indicates that OpenGL is a software interface to graphics hardware.

## المراجع

• Bôcher, Maxime (1907). Introduction to Higher Algebra. Macmillan. pp. 11ff.
• Briot, Charles; Bouquet, Jean Claude (1896). Elements of Analytical Geometry of Two Dimensions. trans. J.H. Boyd. Werner school book company. p. 380.
• Cox, David A.; Little, John B.; O'Shea, Donal (2007). Ideals, Varieties, and Algorithms. Springer. p. 357. ISBN 0-387-35650-9.
• Garner, Lynn E. (1981), An Outline of Projective Geometry, North Holland, ISBN 0-444-00423-8
• Jones, Alfred Clement (1912). An Introduction to Algebraical Geometry. Clarendon.
• Miranda, Rick (1995). Algebraic Curves and Riemann Surfaces. AMS Bookstore. p. 13. ISBN 0-8218-0268-2.
• Wilczynski, Ernest Julius (1906). Projective Differential Geometry of Curves and Ruled Surfaces. B.G. Teubner.
• Woods, Frederick S. (1922). Higher Geometry. Ginn and Co. pp. 27ff.

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## وصلات خارجية

• Jules Bloomenthal and Jon Rokne, Homogeneous coordinates [1]
• Ching-Kuang Shene, Homogeneous coordinates [2]
• Wolfram MathWorld