Felsner S.'s 3-Interval irreducible partially ordered sets PDF

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Sample text

Hence follows this very simple construction* for an ellipsoid (with three unequal axes), by means of a sphere and an external point, to which the author was led by the foregoing process, but which may also be deduced from principles more generally known. From a fixed point a on the surface of a sphere, draw a variable chord ad; let d be the second point of intersection of the spheric surface with the secant bd, drawn to the variable extremity d of this chord ad from a fixed external point b; take the radius vector ae equal in length to bd , and in direction either coincident with, or opposite to, the chord ad; the locus of the point e, thus constructed, will be an ellipsoid, which will pass through the point b.

Where U is (as in art. 19) the characteristic of the operation of taking the versor of a quaternion, or of a vector; and t is a scalar coefficient. Again, the equation 0 = S . ), which expresses that the three vectors ν, µ, κ are coplanar, shows also that the two vectors V . νµ and V . νκ are parallel to each other, as being both perpendicular to that common plane to which ν, µ and κ are parallel; hence we have the following equation between two versors of vectors, or between two vector-units, UV .

At least if we change (for greater facility of comparison of the results among themselves) the ambiguous sign ± to its opposite. We may also suppress the scalar coefficient t, if we only wish to form an expression for a line τ which shall have the required direction of a tangent, without obliging the length of this line τ to take any previously chosen value. The formula for the system of the two tangents to the two lines of curvature thus takes the simplified form: τ = UV . νι ∓ UV . ) in which the two terms connected by the sign ∓ are two vector-units, in the respective directions of the traces of the two cyclic planes upon the tangent plane.

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3-Interval irreducible partially ordered sets by Felsner S.


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