JPS60110013A - Curved surface interpolation method - Google Patents

Abstract

PURPOSE:To attain the following to an optional curve given on an x-y plane through a simple constitution and in real time in case a curved surface shown by parameters (u) and (v) is cut. CONSTITUTION:An arithmetic part 1 which delivers partial differential coefficients of (u) and (v) directions of a curved surface V (u, v) feeding parameter (u) and (v) performs the analyzing differential calculation of the surface V if possible and otherwise calculates a differential coefficient in terms of numerical calculation when said differential calculation is possible. An arithmetic part 2 calculates approximately the minute variation degrees DELTAu and DELTAv of the parameters (u) and (v) of the surface V (u, v) corresponding to the minute shift amounts DELTAx and DELTAy of the X and Y directions. An addition part 3 adds the present parameter values (u) and (v) to the changes DELTAu and DELTAv of parameters to the variation degrees DELTAx and DELTAy. Then an output part 4 delivers the coordinate value V (u+DELTAu, v+DELTAv) on the corresponding curved surface when the variables (u) and (v) are supplied.

Description

[Detailed Description of the Invention] [Technical Field of the Invention] The present invention relates to a curved surface interpolation method for approximating a curved surface expressed by parametric variables with minute line segments,
For example, the present invention relates to a curved surface interpolation method that outputs x and y coordinate command values along a defined arbitrary curve.

[Prior art]

In general, parametric representation is convenient as a method for representing curved surfaces. As shown in FIG. 1, any point P within the area 0≦U≦1, 0≦V≦1 on the two-dimensional plane uv.
By mapping (u, v) to the three-dimensional space, the point P (X r 3' r z
). That is, the area shown in FIG. 1 is mapped to the turn (S) shown in FIG. Here, the point Pt on the curved surface S in the three-dimensional space can be expressed as follows.

V(u・v)=(x(u,v),y(u,v),z(
u, v))・・・・・・・・・(2) When actually cutting the curved surface (S)′ using a numerically controlled machine tool, since the tip of the tool is spherical, the curved surface S Any point above e)
Then, set the modulus 1(N)'r, and set the tool center at a position separated by the tool radius in the normal direction.

The normal (good) is calculated from the cross product operation. Therefore, if the coordinate values on the curved surface S are given, the tool trajectory can be obtained relatively easily.

FIG. 3 is for explaining a conventional curved surface cutting method. The curved surface (S) has a parameter U in three-dimensional space.

It is assumed that the above equation (1) or (2) is defined in V. The cutting tool ('r) is the initial point (P8)
The point (P8) is sufficiently far away from the curved surface, and the point (P8) is moved in fast forward motion to point 9 (Po).

Next, the tool σ) cuts into the curved surface using a cutting feed from the point (Po) to the point (P+)t. This point (Pl) is calculated using equation (2) as u=0. It can be expressed as a point V(0, vl) of v-tension v1. Furthermore, the tool (T) moves on the curved surface from point P to point P2 by cutting feed, but the coordinate command value to the tool (T) is given by V (u, vl), and the parameter V is It is assumed that vl is a constant value, and the parameter U is changed from 0 to 1.

Next, change the two-axis coordinate values from point (P2) to point (P5), remove the tool from the entire curved surface (S), set only the parameter V from f vl to v2, and set the point ( P6) (v
(1,v2)). Next, the tool is moved by rapid traverse to a very close point (P5) directly above point (P6), and then cut to point (P6) by cutting feed. Next, the parametric variable V is kept constant v2, and is varied from ``el'' to 0, and the tool (force) is moved from the point (P6) to the point (Pl).

Similarly, the tool (1) moves from point (P8) to point (P2O) and cuts the curved surface S in a zigzag pattern. In this way, the conventional cutting method fixed one of the parametric variables and varied the other to obtain a tool trajectory on a curved surface. It is impossible to make the cutting locus (pT) of the tool into an arbitrary shape in the projection onto the surface (5xy). By the way, in order to efficiently cut the entire curved surface, or to specify the entire region and cut only that region, it is necessary to cut the curved surface to follow an arbitrary curve defined in XY 1,000 planes. , there has been a desire for a curved surface interpolation method that allows such cutting. In order to achieve this, the known x and y coordinate values X are used. l
There is a method of combining Yo and equation (1) above, calculating the parametric variables ur ve through repeated calculations, and finding points on the curved surface. However, with this method, it is difficult to obtain a convergent solution for complex curved surfaces. , and even if a solution is obtained, the calculation time is long;
It has been impossible to interpolate a curved surface in real time. [Summary of the Invention] The present invention solves the drawbacks of the conventional method and reduces the parameter value U.

The object of the present invention is to provide a curved surface interpolation method that can perform follow-up control in real time to arbitrary curves given on 17 planes with a simple configuration when cutting a curved surface represented by V.

[Embodiments of the invention]

FIG. 5 is a simplified configuration diagram showing an embodiment of the curved surface interpolation method according to the present invention, in which 1 is the parameter u*
vk large input, U and θvav of the curved surface V(u,v) and partial differential coefficients in the Y direction 5(U,V) and -a,
This is an arithmetic unit that outputs (U, V). This calculation unit 1 is
For example, if the surface V (u, v) can be differentially calculated analytically, all the calculations are performed; otherwise, the differential coefficient is calculated numerically, such as Tiller expansion.

2 is the minute movement amount ΔX, Δ in the X and Y directions from the current coordinate point
When y'c is input, the calculation section uses the differential coefficient of the curved surface which is the output of the calculation section 1, and outputs the changes ΔU and ΔV of u and v with respect to the minute changes ΔX and Δy. Parametric variable u,
av av on the curved surface V(u, v) given by v, its partial differential coefficients −(u, v), −y; (u, v
) can be expressed as follows from equation (1) above.

Therefore, between ΔX, Δy and ΔU, ΔV, (
5) The relationship shown in the equation holds true.

When equation (5) is solved for ΔU and ΔvK, the following equation (6) is obtained.

However, d = 1/((including
is the minute movement amount ΔX, Δ in the X and Y directions using equation (6).
Parameter variable U of curved surface V(u, ri) corresponding to y.

It has a function to approximately calculate the minute changes in mass ΔU and Δvf. 3 are the current parameter values u, y, and ΔX.

Minute change ΔU of the noclameter with respect to the amount of change Δy.

This is an addition unit that adds Δma.

Incidentally, 5th: The structure shown in the figure includes an illustrated slant calculation unit that calculates the coordinate values of the corresponding curved surface from the parameter variables u and v.

Reference numeral 4 denotes an output unit which inputs the corresponding coordinate values on the curved surface when the parametric variables u and V are input.

The operation of the curved surface interpolation method of the present invention configured as described above is explained in the sixth section.
This will be explained below using figures.

A curved surface (S) whose current coordinate point has values of parametric variables u and v
Let it be the upper point (Pl). The coordinate value of point (Pi) is V(u
, S=Cx(u,ma)+y(u+v:Lz(usv):
It is 1. The point (Pi) projected onto the XY plane is (
Ql), then the point (Ql) is [x(u, v), y(u
, v), 0]. On the xy plane, point (Ql
) is the point (Q2), and its slight change in the x and y directions is ΔX.

Suppose that it is given as Δy. At this time, according to the curved surface interpolation method having the configuration shown in FIG. 5, the parameter U.

A minute change in V is obtained, and as a result, the coordinate values of the next point (P2) of the point (Pl) are obtained. Strictly speaking, the projection of point (P2) onto the ). It is clear that if a sequence of points is given one after another on the XY plane, a sequence of points corresponding to the sequence of points can be obtained by repeating the same operation. This will be explained using FIG. The curved surface (S) to be cut is defined in a three-dimensional space, and the region SxY is a projection of the curved surface 0) onto the XY plane. Now, XY
Line segment P on the plane. Pn is defined, and a sequence of points P, .about.Pn-1 that divides the line segment is also defined. Starting point (P
If the parameter values u and V in o) are known, PoP, by giving the amount of change 0 0, we can calculate the minute change (Δ
u o HΔu o ) is calculated, and (U + Δuo, v
+ΔU), a point (p
s,) is obtained.

The tool σ) moves from the point (ps 0) to the point (ps, ),
Cutting curved surfaces. At this time, considering that the point (ps,) on the XY plane, point P, and point Q are almost the same, and giving the amount of change P1P2, the amount of change Δu1 of the parametric variables u and V. Δ
V.

Gamaru. Therefore, the point (P2) on the curved surface corresponding to the point (P2)
, 2) is (u0+Δ10+Δu1 g V0+Δv0
+Δv1) is obtained as a point on a curved surface with a tal parameter. By continuing the above operations, the point sequence P is created. rQ
An approximate curve (L8) on the curved surface (S) corresponding to l + Q2...Qn and the above point sequence is obtained.5° Also, when an arbitrary curve is given as shown in Figure 8, the curve t- Even when approximated as a point sequence like Po-Pn, P0→Q on the XY plane using the method described above.

→...Qi-4-Q, +,...→Q It is possible to obtain a tool trajectory on a curved surface having a series of points.

As described above, the curved surface interpolation method of the present invention, given an arbitrary curve on the , the X and Y coordinate values of the trajectory of the tool on the curved surface S represented by the parametric variables follow the curve.

FIG. 9(a), (>> is a diagram illustrating another embodiment of the curved surface interpolation method according to the present invention. As in the embodiment described above, points on the curved surface are However, depending on the curved surface, as shown in FIG.
6) X+ of the sequence of points on the curved surface claimed when given as
The y coordinate value is the original P, such as pol at l *++ Q7. . . . Compared to P7, the hanging may be far apart due to the accumulation of errors9. This means, for example, P,
This is because when approximated by Q, the change meter is given as P1P2 when requesting the next P2. FIG. 9(b) is for explaining a method for eliminating such a drawback. When obtaining an approximation from point P0 to point P, the amount of change P. Suppose that a point Q is obtained by giving Pl.

The errors at this time are P and Q. Point Q2 for point P2
When calculating , the method described above gives P and P2 as the amount of change, but the errors P and Q.

and give QIP2e. In this way, by correcting the previous error by taking into account the next change, the results shown in Figure 9 (
As shown in b), it can be controlled to follow a given point sequence.

〔Effect of the invention〕

As described above, the curved surface interpolation method according to the present invention includes, in cutting a curved surface expressed by the parametric variables B and v, a first calculation unit that calculates the coordinate values of the corresponding curved surface from the parametric variables U and M; Parameter variable U.

a second calculation unit that divides partial differential values in the U direction and v direction from V; and a minute change ΔU in the number of intermediate customers u and V corresponding to minute changes ΔX and Δy in the X @ and Y axis directions; X from the current coordinate value V(u,v).

For the small change command values ΔX and Δy in the Y-axis direction, the following coordinate values are calculated using the calculation results of the first to third calculation units,
By setting v(u+Δu, V+ΔV), it is possible to interpolate a curved surface at high speed with a simple configuration.

Also, given an arbitrary curve defined on the X, Y plane,
By adding means for taking finite value points of the curve and approximating the curve to a polygonal line, the curved surface interpolation method of the present invention is capable of interpolating X along the arbitrary curve.

Curved surface interpolation that outputs the Y coordinate command value becomes possible.

Furthermore, the previous coordinate value Vs (J 171) and the minute change ΔXIHΔy in the X-Y axis direction, and the current coordinate value vl
The error on the X-Y axis when +1 ("i+11vl+1)" is "i+1 ("1-H1vi+1) "i ("i"1
)=x#Fi+1("i+11vi+1)"-Fl("
By correcting 1lvi)-Δy by taking into account the minute change Δ''1+11ΔFi+1 in the X-Y axis direction when calculating the next coordinate value, it is possible to easily improve the surface interpolation accuracy.

The curved surface interpolation method of the present invention can be applied not only to curved surface cutting using a numerically controlled machine tool but also to a curved surface display device, and a graphic display capable of high-speed display can be realized.

[Brief explanation of the drawing]

Figures 1 and 2 are explanatory diagrams for curved surfaces with parametric variable representation, Figure 3 is an explanatory diagram for conventional curved surface cutting methods, and Figure 4 is an explanatory diagram for the conventional curved surface cutting method.
The figure is an explanatory diagram of a cutting method that is impossible with conventional curved surface cutting methods, FIG. 5 is a block diagram for explaining an embodiment of the curved surface interpolation method according to the present invention, and FIG. 6 is a curved surface interpolation method according to the present invention. An explanatory diagram of the operation of the 711ffia3.
(b+, FIG. 8 is an explanatory diagram of the operation of the curved surface interpolation method according to the present invention, and FIG. 9 (1b) is an explanatory diagram of another embodiment of the curved surface interpolation method according to the present invention. 1.2. ...Arithmetic section, 3...addition section, output section for 4. 1 substitute: Masuo Oiwa (12 members)

Claims (1)

[Claims] (Re) The coordinate values of the three axes x+y+z are expressed as
v), y (u, v) 4 (u, v)), a first calculation unit that calculates the coordinate values V (u, v), and a Partial differential a, V (with one u, , V(u, v)
a second calculation unit that calculates minute changes Δ in the parametric variables u and v corresponding to minute changes ΔX and Δy in the X-axis and Y-axis directions;
It has a third calculation unit that calculates U and ΔV based on the calculation results of the second calculation unit using the following formula %, and -current coordinate value Vo (uo
l vo) = 11”o CubITo) +3
'o(uo * To) r Zo(uo+Vo):)
When small changes ΔX and Δy in the x and y axis directions from
A curved surface interpolation method characterized by setting Vo+Δ→, z(uo+Δu, VQ+Δv)]. (2) Depending on the parametric variables u and v, the coordinate values of the three axes X, Y, and Z are vector V (u, v) = (x (u, v)
), z(u, v)), a first calculation unit calculates the coordinate values V(u, v), and a partial differential of the coordinate values u, V by a −, V( u, v) and B, V (u 1) and the U2 calculation unit □ which calculates the minute changes ΔU, ΔV of the parametric variables u, V corresponding to the minute changes ΔX, Δy in the X-axis and Y-axis directions. of,
Based on the calculation result of the second calculation part, the third calculation part calculates the following formula % formula %, and the current coordinate value vo(uoIv
o) = [Xo (uo, vo), yo (uo, vo), z
o (uo, vo)] minute change ΔX in the X and Y axes Xo
, Δy are given, the following coordinate values are expressed as V(u + Δu, v+ΔV) = Cx(u, +Δu+V6+Δv), y(uo+Δu
, vo+Δv:Lz(uo+ΔB, uol4)), and given an arbitrary curve defined on the X, Y plane, means for approximating the curved line by taking a finite number of points on the curved line. , and the said occasion? Each of the finite line segments constituting fM is defined by the minute change ΔX in the X and Y axis directions,
A curved surface interpolation method characterized by setting as Δy. (3) The coordinate values of the three axes of X, Y, and Z are expressed as vector V(u, s=(X (u r V ) *y
(u r V) r Z (u r V ) In the curved surface expressed as (u r V ), a first calculation unit that calculates the coordinate values V (u, v), and a partial differential of the coordinate values u and v by a θ a, V (u, v) and a, V (u,
v), and minute changes ΔU and ΔV of the parametric variables u and v corresponding to the minute changes ΔX and Δy in the Y-axis and Y-axis directions, based on the calculation results of the second computing unit. ,
The current coordinate value Vo(uol
vo ) = (xo(u61 vo )+yo(uo
X, Y from l vo) + Zo ("o l vo))
When small changes ΔX and Δy in the axial direction are given, the following coordinate values are expressed as
+'Vo+Δv), z(uo+Δu, Vo+ΔV)), and given an arbitrary curve defined on the X, Y plane, a finite number of points of the curve are taken to approximate the curve as a broken line, The amount of change 'i of each of the finite line segments constituting this broken line, the minute changes ΔX, Δy in the X and Y axis directions
The means sets the previous coordinate J[Vi(u
l, vl) and minute changes in the x and y axis directions ΔXi, Δyi
Accordingly, the current coordinate value Vl+1(u+++, vt+1
) error on the X and Y axes XI+1(t11+++
vi++) −xl(ul, vl)−Δx. yt+1(ui++, vt+t)-yt(ui, vi)
A curved surface correction method characterized in that −Δy is corrected by taking into account a minute change in the X and Y axis directions ΔXl+11Δyi+1 when calculating the next coordinate value.