A method for tenon piece processing data recognition

Abstract

The present application relates to the technical field of wood mortise and tenon piece processing, and particularly relates to a method for mortise and tenon piece processing data identification, which comprises the following steps: importing a mortise and tenon piece file into three-dimensional CAM software, obtaining initial data of each face of the mortise and tenon piece and storing the initial data into an array FaceData; obtaining minimum value point PMin and maximum value point PMax of the mortise and tenon piece by establishing an overall outer inclusive rectangle of the mortise and tenon piece; obtaining head hole feature point data; establishing a mortise and tenon piece coordinate system and generating an outer inclusive rectangle of a mortise and tenon piece feature face; wherein the short side direction of the outer inclusive rectangle is the X-axis direction, the long side direction of the outer inclusive rectangle is the Z-axis direction, and the cross-multiplication direction of the Z-axis and the X-axis is the Y-axis direction; and generating mortise and tenon piece feature data through the outer inclusive rectangle of the mortise and tenon piece feature face in step 4. According to the method, the three-dimensional model file can be directly imported to generate the required machine tool processing file, and the processing efficiency of the mortise and tenon piece is greatly improved.

Description

Technical Field

[0001] This invention relates to the field of wood tenon and mortise joint processing technology, and in particular to a method for identifying data in tenon and mortise joint processing. Background Technology

[0002] Mortise and tenon joints are a structural method in traditional Chinese architecture, which primarily uses wood and brick as building materials and a timber frame structure. Constructed from main components such as columns, beams, and purlins, the joints between these components are fitted together with mortise and tenon joints, forming a flexible framework. The mortise and tenon joint is an extremely ingenious invention. This method of component connection makes traditional Chinese timber structures a unique flexible structure that surpasses modern building frames, trunk frames, or steel frames. It can not only withstand greater loads but also allows for a certain degree of deformation. Under seismic loads, this deformation can offset some of the seismic energy, reducing the structure's seismic response.

[0003] Tenon (such as) Figure 2 As shown, the core components of mortise and tenon joints mainly include straight tenons, beveled tenons, hook tenons, palm tenons, forked tenons, swivel tenons, and holes. To improve the processing efficiency of tenons, CNC machining is currently used to replace traditional manual operations. This involves three-dimensional data recognition of the tenons to generate feature data suitable for machine tool processing. However, existing 3D CAM software generates overall machining data for the parts, and the generated data is generally difficult to adapt to the CNC machine tool instructions of most manufacturers.

[0004] Therefore, a new technical solution is urgently needed to solve the above-mentioned technical problems. Summary of the Invention

[0005] The purpose of this invention is to overcome the problems of the prior art and provide a method for identifying mortise and tenon machining data. By importing a 3D mortise and tenon file into the system, the machining data required by the machine tool can be generated. The difficulty of the algorithm lies in distinguishing between different mortise and tenon parts. For common mortise and tenon parts such as straight tenons, beveled tenons, and holes, only general feature points such as head features and hole features need to be obtained. However, for irregularly shaped mortise and tenon parts such as tenons with grooves and tenons with loops, specific tenon and loop-shaped tenon identification functions are required for feature determination. This effectively solves the problem that the generated data is generally difficult to adapt to the CNC machine tool instructions of general manufacturers.

[0006] The above objectives are achieved through the following technical solutions:

[0007] A method for identifying tenon machining data includes:

[0008] Step 1: Import the tenon file into the 3D CAM software, obtain the initial data of each face of the tenon and store it in the array FaceData;

[0009] Step 2: By establishing the overall outer rectangle of the tenon, the minimum point PMin and maximum point PMax of the tenon are obtained;

[0010] Step 3: Obtain head hole feature point data;

[0011] Step 4: Establish the coordinate system of the tenon and generate the rectangle enclosing the feature surface of the tenon; wherein, the direction of the short side of the enclosing rectangle is the X-axis direction, the direction of the long side of the enclosing rectangle is the Z-axis direction, and the cross product of the Z-axis and the X-axis is the Y-axis direction;

[0012] Step 5 generates tenon feature data by using the rectangle enclosing the tenon feature surface described in Step 4.

[0013] Furthermore, the initial data mentioned in step 1 includes: topological surface, surface type, surface normal segment, and four extreme points of the bounding rectangle of the surface, wherein the surface normal segment must intersect with the surface.

[0014] Furthermore, step 3 specifically includes the following steps:

[0015] Step 3.1 Store all faces that do not intersect with other faces into the array NoIntFaces;

[0016] Step 3.2 Traverse NoIntFaces; if the number of loops constituting the feature surface is 1, continue.

[0017] Step 3.3 If the normal direction of feature surface 1 is the same as the normal direction of feature surface 2 and they are not collinear, then continue;

[0018] Step 3.4 Use the bool value IsHeadHole to determine whether the feature point is a head or a hole. If the normal direction of feature surface 1 is opposite to the projection direction of the normal point of feature surface 1 onto feature surface 2, then it is a head; otherwise, it is a hole.

[0019] Step 3.5 Determine the shape of the feature surface using the Int value FaceShapel, where 1 represents a full circle, 2 represents a triangle, 3 represents a rectangle, and 4 represents a runway arc.

[0020] Step 3.6 Determine the shape FaceShape2 of feature surface 2 using the same method. When at least one of FaceShape1 and FaceShape2 is not 0, the feature surface is the processing feature surface to be found. Store all processing feature surfaces in the array HeadHoleTenons.

[0021] Further, step 3.1 specifically involves triangulating the topological surface, thus transforming the intersection of a line and the surface into the intersection of a line and a triangle; it also includes:

[0022] Step 3.1.1 Determine whether the face containing the triangle is parallel to the line containing the line segment. If yes, exit; otherwise, continue to step 3.1.2.

[0023] Step 3.1.2 Determine whether the intersection point of the line segment and the face containing the triangle is on the line segment. If yes, continue to step 3.1.3; otherwise, return to flash.

[0024] Step 3.1.2 Determine whether the intersection point is inside the triangle. If so, it means that they intersect.

[0025] Further, step 3.1.2 specifically involves: assuming point M is the intersection of the plane and the line containing pq, and both point M and point a are points on the plane, then:

[0026] (op-t*qp)*normal=oa*normal

[0027] Where o is the origin of the coordinate system, p and q are the two endpoints of the line segment, a, b, and c are the three vertices of the triangle, and normal is the intermediate vector of the plane equation, which will be canceled out later; that is, op is the direction vector from point o to point p, qp is the direction vector from point q to point p, oa is the direction vector from point o to point a, ob is the direction vector from point o to point b, oc is the direction vector from point o to point c, ap is the direction vector from point a to point p, ac is the direction vector from point a to point c, and ab is the direction vector from point a to point b; for any point M on vector pq, pM = t * pq, where t is the ratio of vector pM to pq;

[0028] Solving for t, we get:

[0029] t = (ap * normal) / (qp * normal)

[0030] Determine if t is between (0, 1). If yes, continue to step 3.1.3; otherwise, return false.

[0031] Further, step 3.1.3 specifically includes:

[0032] Point M inside triangle ABC can be represented using the centroid coordinate system of the triangle, then:

[0033] M=(1-lamda2-lamda3)*a+lamda2*b+lamda3*c

[0034] Where a, b, and c are the coordinates of the three vertices of the triangle. If both lambda2 and lambda3 are between (0, 1), it means that M is inside triangle abc. According to the value of t obtained in step 5, the coordinates of the intersection point M of the plane and the line can be calculated by the expression op-t*qp. The values ​​of lambda2 and lambda3 can be obtained by applying Cramer's rule and the rule of operation of the mixed product.

[0035] Furthermore, step 5 specifically includes:

[0036] Step 5.1 Obtain the minimum point P1 and maximum point P2 of the bounding rectangle of the feature surface;

[0037] Step 5.2 Project the maximum point P2 onto a straight line with the minimum point P1 as the center and the X-axis as the direction. The projection point is P_T. Let L1 be the distance from P1 to P_T.

[0038] Step 5.3 Project the maximum point P2 onto a straight line centered at the minimum point P1 with the Z-axis as the direction. Let the projected point be P2_T. Let L2 be the distance from P1 to P2_T. Then:

[0039] The material width is:

[0040] Width = max(L1, L2)

[0041] The material thickness is:

[0042] Thick = min(L1, L2)

[0043] Where max(a, b) finds the larger of a and b, and min(a, b) finds the smaller of a and b;

[0044] Step 5.4 Project point PMin onto a straight line with point P1 as the center and the Z-axis as the direction. The projection point is P3_T. Obtain two auxiliary projection points M1_T and M2_T.

[0045] Where M1_T is the projection point of point P1 onto the straight line with point P2 as the center and the Z-axis as the direction, and M2_T is the projection point of point P2 onto the straight line with point P1 as the center and the Z-axis as the direction.

[0046] If the angle between the direction from the midpoint of P1 and P2 to P3_T and the Z-axis is acute, then the margin_1 of edge 1 is:

[0047] Margin_1 = min(distance from P1 to P3_T, distance from M2_T to P3_T);

[0048] Replace P3_T with the projection point of point PMax onto the straight line centered at point P2 with the Z-axis as the direction. The margin_2 of the second side is:

[0049] Margin_2 = min(distance from M1_T to P3_T, distance from P2 to P3_T);

[0050] Step 5.5 Project point PMin onto a straight line with point P1 as the center and the X-axis as the direction. The projection point is P4_T. Obtain two auxiliary projection points M3_T and M4_T.

[0051] Where M3_T is the projection point of point P1 onto the straight line with point P2 as the center and the X-axis as the direction, and M4_T is the projection point of point P2 onto the straight line with point P1 as the center and the X-axis as the direction.

[0052] If the angle between the direction from the midpoint of P1P2 to P4_T and the X-axis is acute, then the margin_5 for edge number 5 is:

[0053] Margin_5 = min(distance from P1 to P4_T, distance from M4_T to P4_T);

[0054] Replace P3_T with the projection point of point PMax onto the straight line centered at point P2 with the X-axis as the direction. The margin_6 of the 6th edge is:

[0055] Margin_6 = min(distance from M3_T to P4_T, distance from P2 to P4_T).

[0056] Beneficial effects

[0057] The method for identifying tenon machining data provided by this invention has the following advantages:

[0058] 1. The 3D feature data recognition algorithm can directly generate the machine tool processing files required by the user after importing the 3D model file, which greatly improves the processing efficiency of mortise and tenon parts;

[0059] 2. After importing the 3D tenon and mortise file, there is no need to establish a coordinate system for the workpiece as in traditional CAM. The establishment of the coordinate system is completed within the algorithm, further improving efficiency.

[0060] 3. Imported 3D tenon and mortise files do not need to be classified. The classification of tenons and mortises is completed internally by the algorithm, and the processing data corresponding to each tenon and mortise is output. Attached Figure Description

[0061] Figure 1 This is a flowchart of a method for identifying tenon machining data according to the present invention;

[0062] Figure 2 This is a schematic diagram of the tenon structure in the method for identifying tenon processing data according to the present invention. Detailed Implementation

[0063] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments. The described embodiments are merely some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0064] like Figure 1 As shown, a method for identifying tenon machining data includes the following steps:

[0065] S101 Obtains initial topological surface data;

[0066] S102 obtains the maximum and minimum points of the tenon;

[0067] S103 generates the machining feature surface;

[0068] S104 Workpiece coordinate system establishment: If successful, continue; otherwise, terminate.

[0069] S105 generates tenon feature data.

[0070] Specifically, the steps in this embodiment are as follows:

[0071] Step 1: Import the tenon file into the 3D CAM software, obtain the initial data of each face of the tenon and store it in the array FaceData;

[0072] Step 2: By establishing the overall outer rectangle of the tenon, the minimum point PMin and maximum point PMax of the tenon are obtained;

[0073] Step 3: Obtain head hole feature point data;

[0074] Step 4: Establish the coordinate system of the tenon and generate the rectangle enclosing the feature surface of the tenon; wherein, the direction of the short side of the enclosing rectangle is the X-axis direction, the direction of the long side of the enclosing rectangle is the Z-axis direction, and the cross product of the Z-axis and the X-axis is the Y-axis direction;

[0075] Step 5 generates tenon feature data by using the rectangle enclosing the tenon feature surface described in Step 4.

[0076] The initial data in step 1 includes: topological surface, surface type, surface normal segment, and four extreme points of the bounding rectangle of the surface. The surface normal segment must intersect with the surface.

[0077] Step 3 in this embodiment specifically includes the following steps:

[0078] Step 3.1 Store all faces that do not intersect with other faces into the array NoIntFaces;

[0079] Step 3.2 Traverse NoIntFaces; if the number of loops constituting the feature surface is 1, continue.

[0080] Step 3.3 If the normal direction of feature surface 1 is the same as the normal direction of feature surface 2 and they are not collinear, then continue;

[0081] Step 3.4 Use the bool value IsHeadHole to determine whether the feature point is a head or a hole. If the normal direction of feature surface 1 is opposite to the projection direction of the normal point of feature surface 1 onto feature surface 2, then it is a head; otherwise, it is a hole.

[0082] Step 3.5 Determine the shape of the feature surface using the Int value FaceShapel, where 1 represents a full circle, 2 represents a triangle, 3 represents a rectangle, and 4 represents a runway arc.

[0083] Step 3.6 Determine the shape FaceShape2 of feature surface 2 using the same method. When at least one of FaceShape1 and FaceShape2 is not 0, the feature surface is the processing feature surface to be found. Store all processing feature surfaces in the array HeadHoleTenons.

[0084] Specifically, step 3.1 involves triangulating the topological surface, thus transforming the intersection of a line and the surface into the intersection of a line and a triangle; it also includes:

[0085] Step 3.1.1 Determine whether the face containing the triangle is parallel to the line containing the line segment. If yes, exit; otherwise, continue to step 3.1.2.

[0086] Step 3.1.2 Determine whether the intersection point of the line segment and the face containing the triangle is on the line segment. If yes, continue to step 3.1.3; otherwise, return to flash.

[0087] Step 3.1.2 Determine whether the intersection point is inside the triangle. If so, it means that they intersect.

[0088] Specifically, step 3.1.2 involves assuming that point M is the intersection of the plane and the line containing pq, and that both point M and point a are points on the plane, then:

[0089] (op-t*qp)*normal=oa*normal

[0090] Where o is the origin of the coordinate system, p and q are the two endpoints of the line segment, a, b, and c are the three vertices of the triangle, and normal is the intermediate vector of the plane equation, which will be canceled out later; that is, op is the direction vector from point o to point p, qp is the direction vector from point q to point p, oa is the direction vector from point o to point a, ob is the direction vector from point o to point b, oc is the direction vector from point o to point c, ap is the direction vector from point a to point p, ac is the direction vector from point a to point c, and ab is the direction vector from point a to point b; for any point M on vector pq, pM = t * pq, where t is the ratio of vector pM to pq;

[0091] Solving for t, we get:

[0092] t = (ap * normal) / (qp * normal)

[0093] Determine if t is between (0, 1). If yes, continue to step 3.1.3; otherwise, return false.

[0094] Further, step 3.1.3 specifically includes:

[0095] Point M inside triangle ABC can be represented using the centroid coordinate system of the triangle, then:

[0096] M=(1-lamda2-lamda3)*a+lamda2*b+lamda3*c

[0097] Where a, b, and c are the coordinates of the three vertices of the triangle. If both lambda2 and lambda3 are between (0, 1), it means that M is inside triangle abc. According to the value of t obtained in step 5, the coordinates of the intersection point M of the plane and the line can be calculated by the expression op-t*qp. The values ​​of lambda2 and lambda3 can be obtained by applying Cramer's rule and the rule of operation of the mixed product.

[0098] In this embodiment, step 5 specifically refers to:

[0099] Step 5.1 Obtain the minimum point P1 and maximum point P2 of the bounding rectangle of the feature surface;

[0100] Step 5.2 Project the maximum point P2 onto a straight line with the minimum point P1 as the center and the X-axis as the direction. The projection point is P_T. Let L1 be the distance from P1 to P_T.

[0101] Step 5.3 Project the maximum point P2 onto a straight line centered at the minimum point P1 with the Z-axis as the direction. Let the projected point be P2_T. Let L2 be the distance from P1 to P2_T. Then:

[0102] The material width is:

[0103] Width = max(L1, L2)

[0104] The material thickness is:

[0105] Thick = min(L1, L2)

[0106] Where max(a, b) finds the larger of a and b, and min(a, b) finds the smaller of a and b;

[0107] Step 5.4 Project point PMin onto a straight line with point P1 as the center and the Z-axis as the direction. The projection point is P3_T. Obtain two auxiliary projection points M1_T and M2_T.

[0108] Where M1_T is the projection point of point P1 onto the straight line with point P2 as the center and the Z-axis as the direction, and M2_T is the projection point of point P2 onto the straight line with point P1 as the center and the Z-axis as the direction.

[0109] If the angle between the direction from the midpoint of P1 and P2 to P3_T and the Z-axis is acute, then the margin_1 of edge 1 is:

[0110] Margin_1 = min(distance from P1 to P3_T, distance from M2_T to P3_T);

[0111] Replace P3_T with the projection point of point PMax onto the straight line centered at point P2 with the Z-axis as the direction. The margin_2 of the second side is:

[0112] Margin_2 = min(distance from M1_T to P3_T, distance from P2 to P3_T);

[0113] Step 5.5 Project point PMin onto a straight line with point P1 as the center and the X-axis as the direction. The projection point is P4_T. Obtain two auxiliary projection points M3_T and M4_T.

[0114] Where M3_T is the projection point of point P1 onto the straight line with point P2 as the center and the X-axis as the direction, and M4_T is the projection point of point P2 onto the straight line with point P1 as the center and the X-axis as the direction.

[0115] If the angle between the direction from the midpoint of P1P2 to P4_T and the X-axis is acute, then the margin_5 for edge number 5 is:

[0116] Margin_5 = min(distance from P1 to P4_T, distance from M4_T to P4_T);

[0117] Replace P3_T with the projection point of point PMax onto the straight line centered at point P2 with the X-axis as the direction. The margin_6 of the 6th edge is:

[0118] Margin_6 = min(distance from M3_T to P4_T, distance from P2 to P4_T).

[0119] The above description is merely illustrative of the embodiments of the present invention and is not intended to limit the present invention. For those skilled in the art, any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for tenon piece processing data recognition, characterized in that, The application relates to a method for automatically obtaining machining features of a tenon, comprising the following steps: Step 1: importing a tenon file into three-dimensional CAM software, obtaining initial data of each face of the tenon and storing the initial data into an array FaceData; Step 2: obtaining minimum point PMin and maximum point PMax of the tenon by establishing an overall outer inclusive rectangle of the tenon; Step 3: obtaining head hole feature point data; Step 4: establishing a tenon coordinate system and generating an outer inclusive rectangle of a tenon feature surface; wherein the short side direction of the outer inclusive rectangle is the X-axis direction, the long side direction of the outer inclusive rectangle is the Z-axis direction, and the cross direction of the Z-axis and the X-axis is the Y-axis direction; Step 5: generating tenon feature data through the outer inclusive rectangle of the tenon feature surface in step 4; The initial data in step 1 comprises a topological surface, a type of the surface, a normal line segment of the surface, and four extreme points of an outer inclusive rectangle of the surface, wherein the normal line segment of the surface is required to be intersected with the surface; The step 3 specifically comprises the following steps: Step 3.1: storing all surfaces not intersected with other surfaces into an array NoIntFaces; Step 3.2: traversing NoIntFaces, if the number of ring lines constituting a feature surface is one, the step 3.2 continues; Step 3.3: if the normal direction of the feature surface 1 is the same as the normal direction of the feature surface 2 and the two are not collinear, the step 3.3 continues; Step 3.4: determining whether a feature point is a head or a hole by using a bool value IsHeadHole, if the normal direction of the feature surface 1 is opposite to the projection direction of the feature point of the feature surface 1 to the feature surface 2, the feature point is a head, otherwise, the feature point is a hole; Step 3.5: determining the shape of the feature surface by using an Int value FaceShape1, wherein 1 represents a whole circle, 2 represents a triangle, 3 represents a rectangle, and 4 represents a runway circular arc; Step 3.6: determining the shape of the feature surface 2 by using the same method, FaceShape2, when at least one of FaceShape1 and FaceShape2 is not 0, the feature surface is a machining feature surface to be searched, and all machining feature surfaces are stored into an array HeadHoleTenons; The step 5 specifically comprises: Step 5.1: obtaining minimum point P1 and maximum point P2 of the outer inclusive rectangle of the feature surface; Step 5.2: projecting the maximum point P2 to a straight line with the minimum point P1 as a center point and the X-axis as a direction, and obtaining a projection point P_T, and letting L1 be the distance from P1 to P_T; Step 5.3: projecting the maximum point P2 to a straight line with the minimum point P1 as a center point and the Z-axis as a direction, and obtaining a projection point P2_T, and letting L2 be the distance from P1 to P2_T, then: The material width Width is: Width = max (L1, L2) The material thickness Thick is: Thick = min (L1, L2) Wherein max (a, b) is the larger value of a and b, and min (a, b) is the smaller value of a and b; Step 5.4: projecting the point PMin to a straight line with the point P1 as a center point and the Z-axis as a direction, and obtaining a projection point P3_T, and obtaining two auxiliary projection points M1_T and M2_T; Wherein, M1_T is the projection point of point P1 to the straight line with point P2 as the center point and Z axis as the direction, M2_T is the projection point of point P2 to the straight line with point P1 as the center point and Z axis as the direction; If the included angle between the direction of the midpoint of P1 and P2 to P3_T and the Z axis is an acute angle, then the margin of the first side is Margin_1: Margin_1=min (the distance from P1 to P3_T, the distance from M2_T to P3_T); Replace P3_T with the projection point of point PMax to the straight line with point P2 as the center point and Z axis as the direction, and the margin of the second side is Margin_2: Margin_2=min (the distance from M1_T to P3_T, the distance from P2 to P3_T); Step 5.5, project point PMin to the straight line with point P1 as the center point and X axis as the direction, and the projection point is P4_T, and two auxiliary projection points M3_T and M4_T are obtained; Wherein, M3_T is the projection point of point P1 to the straight line with point P2 as the center point and X axis as the direction, and M4_T is the projection point of point P2 to the straight line with point P1 as the center point and X axis as the direction; If the included angle between the direction of the midpoint of P1 and P2 to P4_T and the X axis is an acute angle, then the margin of the fifth side is Margin_5: Margin_5=min (the distance from P1 to P4_T, the distance from M4_T to P4_T); Replace P3_T with the projection point of point PMax to the straight line with point P2 as the center point and X axis as the direction, and the margin of the sixth side is Margin_6: Margin_6=min (the distance from M3_T to P4_T, the distance from P2 to P4_T).

2. A method for tenon processing data recognition according to claim 1, characterized in that, The step 3.1 is specifically to triangulate the topological surface, so that the intersection of the straight line and the surface is converted into the intersection of the straight line and the triangle; further comprising: Step 3.1.1, judge whether the triangle is in the plane and the straight line is parallel, if yes, exit, if not, continue step 3.1.2; Step 3.1.2, judge whether the intersection point of the line segment and the plane is on the line segment, if yes, continue step 3.1.3, if not, return flash; Step 3.1.2, judge whether the intersection point is inside the triangle, if yes, it means intersection.

3. A method for tenon processing data recognition according to claim 2, characterized in that, The step 3.1.2 is specifically: assuming that M point is the intersection point of the plane and the straight line pq, M point and a point are both on the plane, then: (op-t*qp)*normal=oa*normal Wherein, o is the coordinate system origin, p, q are two end points of the line segment, a, b, c are three vertices of the triangle, normal is the intermediate vector of the plane equation, which will be removed subsequently; That is, op is the direction vector of point o to point p, qp is the direction vector of point q to point p, oa is the direction vector of point o to point a, ob is the direction vector of point o to point b, oc is the direction vector of point o to point c, ap is the direction vector of point a to point p, ac is the direction vector of point a to point c, ab is the direction vector of point a to point b; Any point M on the vector pq, pM=t*pq, t is the ratio of vector pM and pq; Solve t is: t= (ap*normal) / (qp*normal) Determine whether t is between (0,1), if yes continue step 3.1.3, if not return false.

4. A method for tenon processing data recognition according to claim 3, characterized in that, The step 3.1.3 is specifically: The point M in the triangle abc can be expressed by the barycentric coordinate system of the triangle, then: M=(1-lamda2-lamda3)*a+lamda2*b+lamda3*c Where a, b, c are three vertices of the triangle, if lamda2 and lamda3 are both between (0,1), then M is inside the triangle abc, according to the value of t obtained in step 5, the intersection point M of the plane and the straight line can be calculated by the expression op-t*qp; The value of lamda2 and lamda3 can be solved by applying Cramer's rule and mixed product operation rule.

Images