Die Geometrie der Lage: Die Geometrie der Lage

Reye, Theodor

Bibliographic data

Multivolume work

Persistent identifier:: 1019887818

Author:: Reye, Theodor

Title:: Die Geometrie der Lage

Sub title:: Vorträge

Year of publication:: 1868

Place of publication:: Hannover

Publisher of the original:: Rümpler

Identifier (digital):: 1019887818

Language:: German

Document type:: Multivolume work

Volume

Persistent identifier:: 1025709268

Author:: Reye, Theodor

Title:: Die Geometrie der Lage

Sub title:: mit einer Aufgaben-Sammlung und einer lith. Figuren-Tafel

Scope:: 1 Online-Ressource (XIV, 268 Seiten, 1 ungezähltes gefaltetes Blatt)

Year of publication:: 1868

Place of publication:: Hannover

Publisher of the original:: Rümpler

Identifier (digital):: 1025709268

Signature of the source:: Mr.II 3000(1/2)

Language:: German

Additional Notes:: Mit einer Aufgabensammlung und einer lithografischen Figurentafel

Usage licence:: Public Domain Mark 1.0

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2018

Document type:: Volume

Collection:: Mathematics

Title page

Document type:: Multivolume work

Structure type:: Title page

quadric ts if, for that esents a the line y j z > O quadric e, to the mch the quation lanes is , honio- quadrie tion for itaining itersect- s points adric in rie. ( )r ave two ; conics, ì points a conic with it. n order 2e ; the power, l of the cripti ve netrical Given , Draw ic. Let onic in te of O Polar plane and tangent plane defined descriptively 27 in regard to A and B. As the line varies in this plane, passing through 0, the locus of P is a line, the polar line of 0 in regard to this conic. Now let another plane be drawn through 0, meeting the former plane in a line l, passing through O, and let the points in which the line l meets the quadric be U and V. If W be the harmonic conjugate of 0 in regard to U and V, the point W is on the polar lines of 0 in regard to both the conics, in which the quadric is met by the two planes drawn through 0; these two polar lines, therefore, intersect one another. Hence, if a third plane, not containing the line l, be drawn through 0, the polar line of 0, in regard to the conic in which this third plane cuts the quadric, will intersect the polar lines of 0 taken in regard to the two former conics, and not pass through the point of intersection of these lines, which is on /. The third polar line will thus lie in the plane of the first two. From this it is clear that, if an arbitrary line be drawn through 0, to meet the quadric in the points Q and R, and P be the point which is the harmonic conjugate of 0 in regard to Q and R, then the locus of P is a plane. This is called the polar plane of 0 in regard to the quadric. From the definition it is manifest that if the polar plane of 0 pass through P, the polar plane of P passes through 0. The tangent plane of the quadric at any point. The polar plane of 0 meets the quadric in a conic. Let II be any point of this. Then the second point, H', in which the line OH meets the quadric, coincides with II; for II is the harmonic conjugate of O in regard to II and II'. We therefore say that the line OH touches the quadric at the point II. Any plane drawn through OH meets the quadric in a conic; this conic is met by OH in two points which coincide at II, that is, HO is the tangent at II of this conic. Again, the conic, also passing through H, in which the quadric is met by the polar plane of 0, has a tangent at II, say t, unless this conic consists of two lines. This line, t, meets the conic in two points which coincide at II, and, therefore, meets the quadric in two points which coincide at II. Consider now the conic in which the quadric is met by the plane containing the two lines HO and t; this conic is met in two points coinciding at II, both by the line HO and by the line t. It cannot, therefore, be a proper conic, but must consist of two lines meeting at II. The plane of these lines is called the tangent plane of the quadric at H. It is to be regarded as the polar plane of the point II; it passes through 0, and the polar plane of 0 passes through II. By choosing 0 suitably, II may be supposed to be any general point of the quadric. But there may be exception if the polar plane of 0 meets the quadric in two lines, as happens when the quadric degenerates into a cone, or into two planes; the polar plane of 0 then passes, respectively, through

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