MATEMATIQKI VESNIK 49 (1997), 77–80

UDK 514.124

originalni nauqni rad research paper

H-PROJECTING IN n-DIMENSIONAL EUCLIDEAN SPACE E^{n}

Milan Janić

Abstract. Several types of projecting in n-dimensional Euclidean spaces are known. In this article we define a new type of projecting of the n-dimensional Euclidean space onto its fixed plane. We shall prove some properties of this projecting. It will be shown that so defined projecting is a central projecting with an (n − 3)-dimensional subspace as a center.

1. Central projecting

By E^{n} we denote the n-dimensional Euclidean space. An m-dimensional subspace of E^{n} will be denoted by E^{m}. It is known that E^{n} can be extended to the projective n-dimensional space P^{n} by adding a hyperplane E_{∞}^{n−1}. The subspaces of P^{n} will be denoted the same way as the subspaces of E^{n}. The lower index ∞ will denote that a subspace of P^{n} is in E_{∞}^{n−1}. If E_{1}^{n} and E_{2}^{n} are subspaces of P^{n}, then their intersection is also a subspace. If n_{1}+n_{2}−n ≥ 0 and the subspaces E_{1}^{n} and E_{2}^{n} are in general position, their intersection is the subspace E^{n_{1}} ∩ E^{n_{2}} = E^{n_{1}+n_{2}−n}.

Let M be a point of E^{n} and {S_{1∞},…, S_{(n−2)∞}} a simplex of a subspace E_{∞}^{n−3}. The points S_{1∞}, …, S_{(n−2)∞}, M determine a subspace E_{M}^{n−2} of E^{n}. Let E_{0}^{2} be a fixed plane of E^{n} in general position with respect to the simplex {S_{1∞},…, S_{(n−2)∞}}, i.e. such that they span E^{n}. We define that E_{0}^{2} ∩ E_{M}^{n−2} = E^{0} = M′, is the projection of the point M by a subspace E_{∞}^{n−3}. The subspace E_{M}^{n−2} is called the projecting subspace. To determine the projection of any other point N onto the plane E_{0}^{2}, it is sufficient to intersect that plane by the subspace determined by the points S_{1∞}, …, S_{(n−2)∞}, N. The subspace E_{∞}^{n−3} is called the center of projecting of E^{n} onto the plane E_{0}^{2}. The point M′ is called the central projection of the point M by the center E_{∞}^{n−3}.

2. Projecting of E^{n} by E_{∞}^{n−3}

Let Ox_{1} … x_{n} be a coordinate system of E^{n}. Let X_{i∞} = x_{i} ∩ E_{∞}^{n−1} (i = 1,…, n) and let E_{1,... ,n−1}^{n−1} be the coordinate hyperplane Ox_{1} … x_{n−1}. If M is a point of E^{n} then its projection onto the hyperplane E_{1,... ,n−1}^{n−1} is M_{1,... ,n−1} = E_{1,... ,n−1}^{n−1} ∩ MX_{n∞}. Let H_{n} be a point of the line X_{1∞}X_{n∞}, H_{n} ≠ X_{1∞}, X_{n∞}. The point M_{1,... ,n−1}^{n} = H_{n}M ∩ X_{1∞}M_{1,... ,n−1} is the projection of M by H_{n} onto the hyperplane E_{1,... ,n−1}^{n−1}. If H_{n−1} is a point of the line X_{1∞}X_{(n−1)∞}, then the procedure of projecting will be continued from the points X_{(n−1)∞} and H_{n−1}.

AMS Subject Classification: 51 N 15

We shall obtain the point M_{1,... ,n−2} = (M_{1,... ,n−1})_{1,... ,n−2} = E_{1,... ,n−2}^{n−2} ∩ M_{1,... ,n−1}X_{(n−1)∞} and the point M_{1,... ,n−2}^{n} = (M_{1,... ,n−1}^{n})_{1,... ,n−2} = E_{1,... ,n−2}^{n−2} ∩ M_{1,... ,n−1}^{n}X_{(n−1)∞} as the projections of the points M_{1,... ,n−1} and M_{1,... ,n−1}^{n} from X_{(n−1)∞} onto the coordinate subspace E_{1,... ,n−2}^{n−2} = Ox_{1} … x_{n−2}. Also M_{1,... ,n−2}^{n−1} = (M_{1,... ,n−1})_{1,... ,n−2}^{n−1} = E_{1,... ,n−2}^{n−2} ∩ H_{(n−1)}M_{1,... ,n−1}, will be the projection of the point M_{1,... ,n−1} from H_{n−1} onto the subspace E_{1,... ,n−2}^{n−2}.

After n − 2 such steps, we shall obtain the set of n − 1 points:

M_{1,2} = ((… ((M_{1,... ,n−1})_{1,... ,n−2}) …)_{1,2,3})_{1,2},

M_{1,2}^{3} = ((… ((M_{1,... ,n−1})_{1,... ,n−2}) …)_{1,2,3})_{1,2}^{3},

M_{1,2}^{4} = ((… ((M_{1,... ,n−1})_{1,... ,n−2}) …)_{1,2,3}^{4})_{1,2},

… M_{1,2}^{n−1} = ((… ((M_{1,... ,n−1})_{1,... ,n−2}^{n−1}) …)_{1,2,3})_{1,2},

M_{1,2}^{n} = ((… ((M_{1,... ,n−1}^{n})_{1,... ,n−2}) …)_{1,2,3})_{1,2}.

H-projecting in n-dimensional Euclidean space E^{n} 79

We call this projecting the H-projecting of E^{n} onto the plane E_{1,2}^{2} = Ox_{1}, x_{2}. We shall prove that the central projecting defined in section 1 is actually an Hprojecting.

Lemma 1. Let E^{n−k−l} be a subspace of E^{n} and let {S_{1∞},…, S_{(k+l)∞}} be a simplex of E_{∞}^{k+l−1} in general position with respect to E^{n−k−l}. Let E_{∞}^{k−1} be the (afine) span of {S_{1∞},…, S_{k∞}}, E_{∞}^{l−1} the span of {S_{(k+1)∞},…, S_{(k+l)∞}}, and E^{n−l} the span of E^{n−k−l} and E_{∞}^{k−1}. If M′ is the central projection of M from E_{∞}^{l−1} onto the subspace E^{n−l}, M′′ the central projection of M′ from E_{∞}^{k−1} onto the subspace E^{n−k−l}, and M′′′ the central projection of M from E_{∞}^{k+l−1} onto the subspace E^{n−k−l}, then M′′′ = M′′.

Proof. Let E^{l} be the span of E_{∞}^{l−1} and M, E^{k+l} the span of E_{∞}^{k+l−1} and M, and E^{k} the span of E_{∞}^{k−1} and M′. Then E^{l} ⊂ E^{k+l} implies M′ ∈ E^{k+l}, which implies E^{k} ⊂ E^{k+l}, and the last relation in turn implies M′′ ∈ E^{k+l} ∩ E^{n−k−l} = {M′′′}.

Theorem 1. The point M_{1,2} defined by the H-projecting is equal to the point M′, defined by the central projecting for S_{i∞} = X_{i∞}, i = 3,…, n, and E_{0}^{2} = Ox_{1}x_{2}.

Proof. We shall prove the theorem by induction. The statement is true for n = 3, we assume it is true for n = m − 1, and let n = m. By Lemma 1 the point M′ is the central projection of the point M_{1,2,... ,m−1} from the span of the points X_{i∞}, i = 3,…, m − 1, onto the plane E_{0}^{2}, which is (by induction) the point M_{1,2}.

We shall also prove the following essential property of the H-projecting.

Theorem 2. The points M_{12}, M_{1,2}^{3},…, M_{1,2}^{n} are on a line parallel to the x_{1} axis.

Proof. The planes which contain the points H_{i} and X_{i∞} do contain the point X_{1∞} ∈ H_{i}X_{i∞}. Hence, 2-dimensional planes spanned by the points H_{i}, X_{i∞} and M_{1,2,... ,i} intersect the coordinate planes E_{1,2,... ,i−1}^{i−1} along the lines which are parallel to the x_{1} axis. Since M_{1,2,... ,i−1}^{i} = E_{1,2,... ,i−1}^{i−1} ∩ M_{1,2,... ,i}H_{i}, and M_{1,2,... ,i−1} = E_{1,2,... ,i−1}^{i−1} ∩ M_{1,2,... ,i}X_{i∞}, we conclude that the lines M_{1,2,... ,i−1}^{i}M_{1,2,...,i−1} are paralel to the x_{1} axis.

By central and H projectings onto the plane Ox_{1}x_{2} the lines parallel to the x_{1} axis remain parallel to it. Hence, the lines M_{12}, M_{1,2}^{i} are parallel to the x_{1} axis, and therefore they coincide, as the central projections of the lines M_{1,2,... ,i−1}^{i}M_{1,2,...,i−1} onto the plane E_{0}^{2}.

By an H-projecting the point M is maped onto an (n − 1)-touple of colinear points M_{12}, M_{1,2}^{3},…, M_{1,2}^{n}. We shall prove that this correspondence is bijective.

Theorem 3. The mappings M ⟶^{H} (M_{1,... ,i}, M_{1,... ,i}^{i+1},…, M_{1,... ,i}^{n}), i =

2, 3,…, n − 1, are bijections of E^{n} onto the set of (n − i + 1)-touples of points (of the coordinate planes E_{1,2,... ,i}^{i}) which are on lines parallel to the x_{1}-axis.

Proof. We need only to prove that given an (n − i + 1)-tuple of points (N_{1},…, N_{n−i+1}) of a line from E_{1,... ,i}^{i} parallel to the x_{1} axis, there is a unique point M such that

(N_{1},…, N_{n−i+1}) = (M_{1,... ,i}, M_{1,... ,i}^{i+1},…, M_{1,... ,i}^{n}).  (1)

The proof will go by induction. Let i = n−1. The point M, if it exists, is on the lines N_{1}X_{n∞} and N_{2}H_{n}. Since N_{1} = N_{2}, or N_{1}N_{2} ∋ X_{1∞} and X_{1∞} ∈ X_{n∞}H_{n}, the lines N_{1}X_{n∞} and N_{2}H_{n} are coplanar nonparallel and intersect at a unique point M.

Assuming the statement is true for i = k ≤ n − 1 we prove it is true for i = k − 1. Let us suppose that (1) holds for some point M and i = k − 1. As we have just proved, the points N_{1} = M_{1,... ,k−1} and N_{2} = M_{1,... ,k−1}^{k} give rise to a unique point N ∈ E_{1,2,... ,k}^{k} such that N = M_{1,2,... ,k}. The point N = M_{1,2,... ,k} in turn determines the line through it parallel to the x_{1} axis which, as we have shown in the proof of the previous theorem, should contain the points M_{1,2,... ,k}^{j}, j = k + 1,…, n. The points M_{1,2,... ,k}^{j}, j = k + 1,…, n, are therefore unique intersections of the line through N = M_{1,2,... ,k} which is parallel to the x_{1} axis, and the lines through the points N_{j−k+2} = M_{1,... ,k−1}^{j}, j = k + 1,…, n, which are parallel to the x_{k} axis. Now, using the induction, we conclude that n − k + 1-tuple (M_{1,... ,k}, M_{1,... ,k}^{k+1},…, M_{1,... ,k}^{n}) uniquely determines M.

Remark. If the coordinate system Ox_{1} … x_{n} is orthogonal, and the directions H_{i}, i = 3,…, n disect the right angles defind by the directions X_{1∞} and X_{i∞}, then M_{1,2}M_{1,2}^{i} equals to the distance of the point M to the hyperplane E_{1,... ,i−1,i+1,... ,n}^{n−1}.

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(received 14.06.1994, in revised form 03.11.1997.)

Technical Faculty, Bor, Yugoslavia