^{1}

^{1}

^{1}

We construct two conical surfaces which take non-coplanar lines as generatrix and rational Bezier curve as ridge-line, and prove that the intersecting line of conical surface has similar properties to Bezier curve. Then, the smoothly blending of two cylinders whose axes are non-coplanar is realized by taking intersecting line of conical surface as axes.

Blending surface is one of the main research fields of curve and surface modeling technology. In recent years, many scholars have focused on algebraic surface blending [

In this paper, we construct equation of ridge-line of conical surface, and blend smoothly two elliptical tubes whose axes are non-coplanar with the elliptical tube that takes intersecting line of conical surface as axes. In the case of dihedral angle of axes line is 90 degree, more than 90, and less than 90, the tube that takes intersecting line of conical surface as axes blend smoothly the two tubes whose axes is non-coplanar. Let l 1 and l 2 be non-coplanar straight line, a is the shortest distance between the two lines, select the x axis so that it is parallel to the shortest distance segment between l 1 and the l 2 , the y axis is parallel to l 1 , and intersects the l 2 . Establishing right hand Cartesian coordinate system, so that the intersection point of l 1 and x axis is point V 0 ( a , 0 , 0 ) , and the intersection point of y axis and l 2 is point V 2 ( 0 , a , 0 ) , the point V 0 ( a , 0 , 0 ) and V 3 ( 0 , c , d ) on l 2 forms four points, these point are no coplanar. Respectively, with V 0 ( a , 0 , 0 ) and V 3 ( 0 , c , d ) as the cone-vertices of the conical surface, and V 1 ( a , a , 0 ) , V 2 ( 0 , a , 0 ) , V 3 ( 0 , c , d ) , and V 0 ( a , a , 0 ) , V 1 ( 0 , a , 0 ) , V 2 ( 0 , c , d ) are the vertices of the characteristic polygons. m 1 , m 2 , m 3 and w 1 , w 2 , w 3 are the corresponding the vertex of weight. The parameter equation for ridge-line of conical surface and equation of conical surface are as follows.

{ x ( u ) = m 1 ⋅ a ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ 0 ⋅ u ( 1 − u ) + m 3 ⋅ 0 ⋅ u 2 m 1 ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ u ( 1 − u ) + m 3 ⋅ u 2 , y ( u ) = m 1 ⋅ a ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ a ⋅ u ( 1 − u ) + m 3 ⋅ c ⋅ u 2 m 1 ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ u ( 1 − u ) + m 3 ⋅ u 2 , z ( u ) = m 1 ⋅ 0 ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ 0 ⋅ u ( 1 − u ) + m 3 ⋅ d ⋅ u 2 m 1 ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ u ( 1 − u ) + m 3 ⋅ u 2 .

{ x ( s ) = w 1 ⋅ a ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ a ⋅ s ( 1 − s ) + w 3 ⋅ 0 ⋅ s 2 w 1 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ s ( 1 − s ) + w 3 ⋅ s 2 , y ( s ) = w 1 ⋅ 0 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ a ⋅ s ( 1 − s ) + w 3 ⋅ a ⋅ u 2 w 1 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ s ( 1 − s ) + w 3 ⋅ s 2 z ( s ) = w 1 ⋅ 0 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ 0 ⋅ s ( 1 − s ) + w 3 ⋅ 0 ⋅ s 2 w 1 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ s ( 1 − s ) + w 3 ⋅ s 2 .

{ x ( u , v ) = a + ( m 1 ⋅ a ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ 0 ⋅ u ( 1 − u ) + m 3 ⋅ 0 ⋅ u 2 m 1 ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ u ( 1 − u ) + m 3 ⋅ u 2 − a ) v , y ( u , v ) = 0 + ( m 1 ⋅ a ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ a ⋅ u ( 1 − u ) + m 3 ⋅ c ⋅ u 2 m 1 ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ u ( 1 − u ) + m 3 ⋅ u 2 − 0 ) v , z ( u , v ) = 0 + ( m 1 ⋅ 0 ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ 0 ⋅ u ( 1 − u ) + m 3 ⋅ d ⋅ u 2 m 1 ⋅ ( 1 − u ) 2 + 2 m 2 ⋅ u ( 1 − u ) + m 3 ⋅ u 2 − 0 ) v . (1)

{ x ( s , t ) = 0 + ( w 1 ⋅ a ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ a ⋅ s ( 1 − s ) + w 3 ⋅ 0 ⋅ s 2 w 1 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ s ( 1 − s ) + w 3 ⋅ s 2 − 0 ) t , y ( u , v ) = c + ( w 1 ⋅ 0 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ a ⋅ s ( 1 − s ) + w 3 ⋅ a ⋅ u 2 w 1 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ s ( 1 − s ) + w 3 ⋅ s 2 − c ) t , z ( u , v ) = d + ( w 1 ⋅ 0 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ 0 ⋅ s ( 1 − s ) + w 3 ⋅ 0 ⋅ u 2 w 1 ⋅ ( 1 − s ) 2 + 2 w 2 ⋅ s ( 1 − s ) + w 3 ⋅ s 2 − d ) t . (2)

{ x ( s ) = − ( 8 w 2 m 2 2 a ( s − 1 ) 2 ( w 1 s − 2 w 2 s − w 1 ) ) / ( m 1 m 3 w 3 2 s 3 − 8 m 2 2 w 1 w 2 s 3 + 16 m 2 2 w 2 2 s 3 − 8 m 2 2 w 2 w 3 s 3 + 24 m 2 2 w 1 w 2 s 2 − 32 m 2 2 w 2 2 s 2 + 8 m 2 2 w 2 w 3 s 2 − 24 m 2 2 w 1 w 2 s + 16 m 2 2 w 2 2 s + 8 m 2 2 w 1 w 2 ) , y ( s ) = ( 16 a m 2 2 w 2 2 s 3 − 8 a m 2 2 w 2 w 3 s 3 + c m 1 m 3 w 3 2 s 3 − 32 a m 2 2 w 2 2 s 2 + 8 a m 2 2 w 2 w 3 s 2 + 16 a m 2 2 w 2 2 s ) / ( m 1 m 3 w 3 2 s 3 − 8 m 2 2 w 1 w 2 s 3 + 16 m 2 2 w 2 2 s 3 − 8 m 2 2 w 2 w 3 s 3 + 24 m 2 2 w 1 w 2 s 2 − 32 m 2 2 w 2 2 s 2 + 8 m 2 2 w 2 w 3 s 2 − 24 m 2 2 w 1 w 2 s + 16 m 2 2 w 2 2 s + 8 m 2 2 w 1 w 2 ) , z ( s ) = ( d m 1 m 2 w 3 2 s 3 ) / ( m 1 m 3 w 3 2 s 3 − 8 m 2 2 w 1 w 2 s 3 + 16 m 2 2 w 2 2 s 3 − 8 m 2 2 w 2 w 3 s 3 + 24 m 2 2 w 1 w 2 s 2 − 32 m 2 2 w 2 2 s 2 + 8 m 2 2 w 2 w 3 s 2 − 24 m 2 2 w 1 w 2 s + 16 m 2 2 w 2 2 s + 8 m 2 2 w 1 w 2 ) . (3)

Theorem 1. The parameter equations of two conical surfaces are (1) and (2). The parameter equation of the intersection line is

r ( s ) = ( x ( s ) , y ( s ) , z ( s ) ) .

Then the first and end points of the curve have the following properties.

r ( 0 ) = V 0 , r ( 1 ) = V 3 .

The nature of the tangent vector is further introduced.

r ' ( 0 ) = 2 w 2 w 1 ( V 1 − V 0 ) , r ' ( 1 ) = 8 m 2 2 w 2 m 1 m 2 w 3 ( V 3 − V 2 ) .

Its geometric meaning is that the first and last points of the curve are tangent to the two different lines.

In this way, we can construct a tube with an intersecting line of conical surface as the axis, and smoothly blend the non-coplanar tubes with two non-coplanar lines as axes.

Theorem 2. The parameter equations of non-coplanar lines l 1 and l 2 are as follows,

l 1 : ( a , a s , 0 ) ; l 2 : ( 0 , c + ( c − a ) s , d s ) .

The curve equation for smoothly blending of two non-coplanar lines is (3), then the parameter equations of two non-coplanar tubes and smoothly blending tube are as follows.

{ x 1 ( s , t ) = x 1 ( s ) + N 11 ( s ) cos t + B 11 ( s ) sin t , y 1 ( s , t ) = y 1 ( s ) + N 12 ( s ) cos t + B 12 ( s ) sin t , z 1 ( s , t ) = z 1 ( s ) + N 13 ( s ) cos t + B 13 ( s ) sin t .

{ x 2 ( s , t ) = x 2 ( s ) + N 21 ( s ) cos t + B 21 ( s ) sin t , y 2 ( s , t ) = y 2 ( s ) + N 22 ( s ) cos t + B 22 ( s ) sin t , z 2 ( s , t ) = z 2 ( s ) + N 23 ( s ) cos t + B 23 ( s ) sin t .

{ x ( s , t ) = x ( s ) + N 1 ( s ) cos t + B 1 ( s ) sin t , y ( s , t ) = y ( s ) + N 2 ( s ) cos t + B 2 ( s ) sin t , z ( s , t ) = z ( s ) + N 3 ( s ) cos t + B 3 ( s ) sin t .

where N = ( N 1 , N 2 , N 3 ) and B = ( B 1 , B 2 , B 3 ) are the unit principal vector and the unit binormal vector at the corresponding point of the intersecting line of conical surface, N i = ( N i 1 , N i 2 , N i 3 ) and B i = ( B i 1 , B i 2 , B i 3 ) , for unit principal vector and unit binormal vector at the intersecting line of conical surface respectively.

In this paper, by constructing the intersecting line of conical surface in

This work is supported by the National Science Fund of China (NSFC11561052) and the Inner Mongolia Natural Science Foundation of China (NMDGP1415).

Bai, G.Z., Wu, Z. and Lin, X. (2017) Intersecting Line of Conical Surface and Smoothly Blending of Two Tubes Whose Axes Are Non-Coplanar. Journal of Applied Mathematics and Physics, 5, 1887-1891. https://doi.org/10.4236/jamp.2017.59158