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Quadrics associated with a simplex in n-space

Published online by Cambridge University Press:  09 April 2009

Augustine O. Konnully
Augustine O. Konnully
Affiliation:
St. Albert's College, Ernakulam Cochin, India
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It is well known that the projections of a pair of points from the vertices of a triangle onto the opposite sides lie on a conic and that when the points are the centroid and orthocentre of the triangle, this conic is a circle. Analogously the projections of the centroid and orthocentre of a simplex from its vertices onto the opposite (n—1)-dimensional faces, if the simplex is orthocentric, lie on a hypersphere [2, 5]. Further the projections of two points onto the edges of a general simplex from the opposite faces lie on quadric [1]; and when the points are the centroid and orthocentre respectively and the simplex is orthocentric, this quadric is a hypersphere [2]. The results as regards projections onto (n—l)-dimensional and 1-dimensional faces being thus known, it remains to see what results hold in the case of intermediary faces. And in this note we prove that a similar result holds for projections onto intermediary faces as well.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 4 , November 1971 , pp. 461 - 465
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Asghar, Hameed, ‘A quadric associated with two points’, Pakistan Journal of Scientific Research 3 (1951), 48–51.Google Scholar
[2]Mandan, S. R., ‘Altitudes of a simplex in n-space’, Jour. Australian Math. Soc. 2 (1962), 403–424.CrossRefGoogle Scholar
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[4]Konnully, Augustine O., Simplexes self-polar for a simplex Jour. Australian Math. Soc. 12 (1971), 309–314.CrossRefGoogle Scholar
[5]Augustine, O. Konnully, Orthocentre of a simplex Jour. Lond. Math. Soc. 39 (1964), 685–691.Google Scholar
[6]Baker, H. F., Principles of Geometry 3 (Cambridge, 1934), 70–72.Google Scholar