5Þ4. Проведем окружность с центром в точке S и радиусом
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAYCAYAAAAPtVbGAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADoSURBVEjHY2hsbGSgNWYYuZYAwX9iMMWW0MUnNLWEHAsGvyX19bGSxgwMd0Fism45xTTzCcgiIwbZUx559YZUsQRbUNXneRjKynqsTOvo4KGZJTlussUMxlELqRYn6JZ0dKTxeMgw7DGObvAB+4iB4Q0sI6LHEfnxATbY6FRsfb0kOBEYR9cT7ROsxQKeoEK2jChLYIYRY0mUMcNCRFllfJdoS3AZjC/pQnwiexNfMibLkoZoYx8GGY89oKQLSwCgyAazPaKTSUpd2OIHFlTIKQgcP2C1sm+w+YgsS6hedtHFktGGBDIGAOfca0c1ftiiAAAAAElFTkSuQmCC)
. Нам дано, что
![](data:image/png;base64,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)
. Но любая окружность, перпендикулярная проведенной и проходящая через точку
М, не лежащую на проведенной окружности, проходит и через точку
М’, мы это показали ранее. Значит, действительно, точки
М и
М’ симметричны в смысле определения 4.
Чтобы это было действительно преобразование, допускают, что точка S отображается в бесконечно удаленную точку, и наоборот (в данном случае нам удобнее мыслить бесконечно удаленную область как одну точку).
Определение 5 менее геометрично, чем предыдущее, но обладает преимуществом большей простоты. Исходя из этого определения, инверсию иногда еще называют преобразованием обратных радиусов. С этим определением связано также название «инверсия» (от латинского слова inversio – обращение).
Очевидно, слова «точка М’ лежит на луче SM и произведение
![](data:image/png;base64,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)
» можно с успехом заменить словами «точки
S,
M и
М’ коллинеарны и скалярное произведение векторов
![](data:image/png;base64,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)
». Здесь
k всегда положительно. Но иногда полезно рассмотреть преобразование, которое переводит точку
M в
М’ так, что
![](data:image/png;base64,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)
и точки
S,
M и
М’ коллинеарны, но
M и
М’ лежат по разные стороны от точки
S. Тогда, очевидно,
k будет отрицательным. Такое преобразование называют инверсией с центром в точке
S и отрицательной степенью. Здесь также допускают, что центр инверсии переходит в бесконечно удаленную область, и наоборот.
Вообще, говоря об инверсии, имеют в виду обычно инверсию с положительной степенью. Если знак степени инверсии может быть любым, то такое преобразование называют обобщенной инверсией. Его определение будет таким.
Определение 6. Обобщенной инверсией плоскости с центром в точке S и степенью инверсии k называется преобразование, которое всякую точку М плоскости, отличную от S, отображает в такую точку М’, что точки S, M и М’ коллинеарны и скалярное произведение векторов
![](data:image/png;base64,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)
. При этом считают, что
S переходит в бесконечно удаленную область, и наоборот.
Это преобразование инволютивное, поскольку точки М и М’ входят в формулу
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGkAAAAXCAYAAAAIqmGLAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAJZSURBVGje7VgxjsIwEPQD+MBJMR0PANNedwoW4gMQUR2iQm54gKFLwwOOjpI/0F7DH/gBf7i7NTgYn2M7YKIURtqGON7Zndn1Omi9XqNozbaYhEhStJeR9Pf7cVmVta49KoNuOL5YSbGSokWSognL83mLJugg2yrJVqOHSNI3QgifKeNdzqdvhGQc1nBGuxihs/rcez8y2YUM9FX4iv8TepjneSskWYsULxEipynnb5VJEoEidMLpYmkKQmd+QtDOlYQLIEiAG5TL6sQnfb2CJIHLQ6zlgA2gRCIw3qvB2pIjbZWRUZIk37BGTexTCqwJX/H+k5VfJjRXq7OTVKK8jNJPNTnQUvr9/he0FRMBsv34qLlam6gPHzwLIS5dGFC1Yz7uikq1HANWln0OtQnBG8qyoVivOQAVDtpkWwAJpMa68YEofBRfWWhXf0CYbX+vRJSBF2vagy3j444pSJaSGSjzohp3Qh8lqm58l6pzXIYtgpQtFPwJ3w7x+gMynAHgANqAaQKSz257+A0M8gCvnLCa8AURmJg6e8eM0SHGdO8aSP69TFI2K518NMYzQrgMTA30bgyWivdsdTaSmoAv3Oh9f3XwJklVl3HCUZSoBqrO/GPOOtDnZXJCtrqm4Hum3UmsEIfvhOc12pruCvphdwkWn2GSUpURspU0HZ/vWdpD+AgYVMK8SSoUoqlAH09N9xGpSNVhceMPdBFsOr4qo7cUhRQeYx/vmEw2VpLU9iADupm2qaGsRcBXJ3eTl2LPtLym43v0K4OPUOIHz/gVPFokKZIUrSn2C/2ApJCoo6r+AAAAAElFTkSuQmCC)
равноправно, а для центра инверсии и бесконечно удаленной области все очевидно.
1.3. Формула инверсии в комплексно сопряженных координатах. Найдем формулу обобщенной инверсии при задании точек комплексными числами. Пусть точкам S, M и М’ соответствуют комплексные числа s, z и z’.
По формуле скалярного произведения векторов
![](data:image/png;base64,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)
. Коллинеарность точек
S,
M и
М’ дает равенство
![](data:image/png;base64,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)
. Отсюда имеем
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIsAAAAYCAYAAADK6w4SAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAJ6SURBVGje7Vo9bsIwFPYBuAASZusBguduVbAq5kohYmrFhLJwAMOWpRdgY+QOnIAbdOAG3KHF+SM1SfycYONIHr4FLPy+5+89f88Cbbdb5OAAgUuCgxOLg0GxsIh6GNPDMo4HNgYex8vBdEx2C8aGsrW2c2nL61H70RE6IoR+OUi4mSmJZeXjNULkbCrgtkhEgNCliaDtXIrDGtFjLmYIr0cDkqe7DzYhmZUDLxZmyquCtrYH2JOxxZAgdK5KbB0XqwSfxS/G2cRLB+YE7RGZ78HXkOkAH4VEFEJV9IVL0VkqDqqKl07BynJ134oaKjFtVdfKvq6JWPDi+9GHdrXfdRV8oRHzqhKO/dXaRi4yHvz7cuxNvHQWW8ACL+lyeYyCgEEKv92j+IcTNHGncrWXD1CWuORAMnHYxAXCgwu3bv8yL61+JcsVF05dLGCzmKrvVg2RT77ECn/mnZp2ijQ+m7lAvEEdL3jXEtCwXy5enqskLw1rhQTWj05Vrt2oU5fsW47fVi4QHk289E2Uk1MY0XfZ84JSUHnFQu7QropXNXkqYlHhYpqHabEU3q3GD7YWS1UL1/+OknoLlaTaxkWFh6pYuoi57J8gExGo0kJCWF4R+Y/qduiqo2+VZ7GBS9cRXuZZusY2QfjEfxsyeUmnobIB6lolqm8PYuAhpZ91dypkGjLNpQ0Pk9OQeDXmniqK3l4xmX8rv7Nw9QUB88rtTvdD1/979AaZYZW9s5jmosrD9DuLOJnlHblOnO4Ft2e8ngnrA+ziCfrGxfbCffrEozuhfeHShw7fHPB4urP5/ywU4wNEBLZzacvLGrE4ODixODixODixOFiEP1UGK3dIxga3AAAAAElFTkSuQmCC)
Û
![](data:image/png;base64,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)
, откуда и получаем искомую формулу
![](data:image/png;base64,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)
.
Итак, обобщенная инверсия имеет формулу
![](data:image/png;base64,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)
или, что то же самое,
![](data:image/png;base64,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)
. При
k>0 получаем инверсию с положительной степенью, при
k<0 – с отрицательной.
Но всякое ли преобразование плоскости, заданное формулой
![](data:image/png;base64,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)
, является обобщенной инверсией? Если принять
, ![](data:image/png;base64,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)
, то достаточно потребовать, чтобы
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
для обобщенной и
![](data:image/png;base64,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)
для обычной инверсии (с положительной степенью).
Значит, всякое преобразование плоскости, задаваемой формулой
![](data:image/png;base64,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)
, есть обобщенная инверсия.
1.4. Неподвижные точки и окружность инверсии. Исследуем уравнение инверсии на неподвижные точки: для них должно выполняться равенство
![](data:image/png;base64,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)
Û
![](data:image/png;base64,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)
. Мы не рассматриваем центр инверсии и бесконечно удаленную область, так как мы доопределили, что они не остаются неподвижными, а переходят друг в друга. Тогда будет выполняться равенство
![](data:image/png;base64,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)
.
Очевидно, что если
![](data:image/png;base64,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)
, то все искомые точки образуют окружность с центром в точке с координатой
sи радиусом
![](data:image/png;base64,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)
. Эта окружность при
![](data:image/png;base64,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)
называется окружностью инверсии. Если обозначить радиус окружности инверсии через
R, то выполняется
![](data:image/png;base64,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)
. И формулу инверсии для
k>0 можно переписать более наглядно:
![](data:image/png;base64,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)
.
Если степень инверсии отрицательна, то преобразование не имеет неподвижных точек (поскольку невозможно изобразить на плоскости, даже комплексной, точки, координаты которых удовлетворяют равенству
![](data:image/png;base64,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)
). Но иногда эту мнимую окружность также называют окружностью инверсии, ее центр расположен в центре инверсии, а радиус будет равен
![](data:image/png;base64,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)
=
![](data:image/png;base64,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)
=
![](data:image/png;base64,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)
.
Так как
![](data:image/png;base64,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)
, то, очевидно, инверсию отрицательной степени легко представить в виде коммутативной композиции инверсии с положительной степенью
![](data:image/png;base64,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)
и центральной симметрии
![](data:image/png;base64,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)
с общим центром в
s.
1.5. Образы прямых и окружностей при обобщенной инверсии.Без ограничения общности рассуждений можно принять
![](data:image/png;base64,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)
, и формула инверсии примет вид
![](data:image/png;base64,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)
, более удобный для практики. Ведь нам пока не важны коэффициенты в получающейся формуле, важно, какую фигуру она описывает.
Пусть задана прямая l с уравнением
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
. При подстановке в это уравнение
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
получаем:
![](data:image/png;base64,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)
. Умножим на
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAQCAYAAAAWGF8bAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAB7SURBVDjLY2hsbGSgJmYYNRC3QgaG/9gwfV3Y0ZHG4yHDsAfVFbJvXPLybLGJe+TVG+I1sD4vUjUyr14Vxo8yZlhoHN3gg0ucpDDEpQmfYTgNJNcwrAZSYhiGgeia6iMjDWPr6yVxieM1MMdNthgjnRlHLcQlPpqX4RgAMPrQqv/f2dYAAAAASUVORK5CYII=)
, это будет равносильным преобразованием, поскольку
![](data:image/png;base64,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)
; получим, опуская в полученном результате штрихи:
![](data:image/png;base64,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)
.
Если q= 0, то получаем уравнение
![](data:image/png;base64,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)
. Так как
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACsAAAAUCAYAAAAUccS4AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFGSURBVEjH7VZLDoIwEJ0DcAET5xDQtTtDGuMFhLhlRbrhAJUdGw/h0jPoCbgDN+AOalsbAcunEI0kLmZDOp333rzOAGmawlxiNkDnDTbLIocu4QoANxEkPGx/XtnYxwSAFHvOF98Cwvl+QQAKJZS5tjExIHACEpy+DVR38hCSrQlwb+LUkF3qIS7FWdJrlGVOzY6NvLdEzWrHd+6rLY8YobQg7oF36bKTFgf9OOmzYqcSAvgUhYViffmcURcByuY5JRqWlHG3cxqIRHl4oJrq4mcHOsIEXOeawda/G1h6ecjoBpGetYc+peoksMonWol6C2zCpittNlBYWmygLSCMbjsRpthg1ANTLxdzwaQK/JOqjh5dzUEsmT0uYGy9QhIchxSVRRDPtvZpWqFtg7ZuLX1BlfGQBTCmG7V6tuv2/4v4B/vDcQc56+L5TW1a+wAAAABJRU5ErkJggg==)
, то умножим обе части уравнения на
![](data:image/png;base64,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)
, получим
![](data:image/png;base64,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)
. Это уравнение прямой, совпадающей с заданной прямой
l. Если
![](data:image/png;base64,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)
, то получаем уравнение окружности
![](data:image/png;base64,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)
, так как
![](data:image/png;base64,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)
. Она содержит центр инверсии, ее центр расположен в точке
![](data:image/png;base64,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)
, а радиус равен
![](data:image/png;base64,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)
. Заметим, что центр лежит на прямой
![](data:image/png;base64,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)
, проходящей через центр инверсии перпендикулярно
l.