Chen Zhanqing

HomeWork 2Solution

1What are the two main applications of a surface modeling system in a product cycle

Answer Generation of visual model for aesthetic evaluationGeneration of NC tool paths for machining

2 Give at least two types of primitives in solid modeling that can also be defined as sweeps

Answer CylinderTorus

3Suppose a solid is stored in computer with CSG data structure When it is displayed on the screen computer needs its faces to triangulate them and render those triangles on the screen Does CSG have those faces Why If not how does the computer get those faces What is the main disadvantage of this What is the main advantage of CSG

Answer CSG does not have those faces because it only stores the history of applying Boolean operations on the primitives It is necessary to transform the CSG into other types of representation

Answer

Advantage The CSG tree data structure has the following advantages1048698 Data structure is simple and stores compact data Accordingly the management of data iseasy1048698 The solid stored in a CSG tree is always a valid solid A valid solid implies a solid of whichinside and outside region can be clearly identified An example of an invalid solid would be asolid with a strut edge In this case the notion of inside and outside is not clear around thevertex where the strut edge is attached1048698 CSG representation of a solid can be always converted to the corresponding B-rep Thus CSGtree representation can be interfaced to the application programs written for B-Rep1048698 Parametric modeling can be realized very easily by changing the parameter values of theassociated primitives

DisadvantageHowever the CSG tree structure has the following disadvantages1048698 Since CSG tree structure stores the history of applying the Boolean operations only theBoolean operations are allowed in the modeling process With Boolean operations alone therange of shapes to be modeled is restricted severely Furthermore the convenient localmodification functions such as lifting and rounding can not be used1048698 It requires a large amount of computation to derive the information on the boundary surfacestheir boundary edges and the connectivity between these boundary entities from CSG treerepresentation Unfortunately there are many applications which require these boundaryinformation One example would be a display of solids Whether a shaded image or a linedrawing of a solid is to be displayed as explained in Chapter 3 the information on the facesandor edges is required Thus CSG tree representation has been assumed to be inappropriatefor the interactive display and manipulation of solids Another example would be thecalculation of NC tool paths to machine the surfaces of a solid by a milling machine In thisapplication the information on the surface to be machined and its boundary edges isnecessary Furthermore the adjacent surfaces need to be derived for the gouging detection Itis not a simple task to derive all these boundary information from a CSG tree representation ofa solid

4(1)During building the Oct-tree representation of a solid when deciding the color of an octant (ie whether it is inside outside or intersecting the boundary of the object) you can not simply decide it by merely looking at the center point of the octant why (2)Then how do you decide the color of the octant Whatrsquos the major disadvantage of it

Answer (1)Because only decide the center point the area that inside or outside can not be known clearly This cause large error To decide the color of the Octant in the following way Step 1Make a hexahedron H that completely encloses the solidStep 2Divide H into 8 octantsStep 3For each octant obtained in step 2 mark it by one of three colorsWhite completely outside the solidBlack completely inside the solidGray partially inside the solidStep 4For each gray octant obtained in step 3 go to step 2 until either no gray octants are left or the size of the octant reaches a minimum threshold(2)The major disadvantage is time consuming

5 Euler-Poincare formula of solids with voids (1) What is the relationship between V E F H and P of the solid below (The small cube is a hollow part also called a void)(2) Let W be the number of voids in the solid establish the relationship between V E F H P and W (Hint first exam the case of two voids and compare with that of a single void)

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids

1What are the two main applications of a surface modeling system in a product cycle

Answer Generation of visual model for aesthetic evaluationGeneration of NC tool paths for machining

2 Give at least two types of primitives in solid modeling that can also be defined as sweeps

Answer CylinderTorus

3Suppose a solid is stored in computer with CSG data structure When it is displayed on the screen computer needs its faces to triangulate them and render those triangles on the screen Does CSG have those faces Why If not how does the computer get those faces What is the main disadvantage of this What is the main advantage of CSG

Answer CSG does not have those faces because it only stores the history of applying Boolean operations on the primitives It is necessary to transform the CSG into other types of representation

Answer

Advantage The CSG tree data structure has the following advantages1048698 Data structure is simple and stores compact data Accordingly the management of data iseasy1048698 The solid stored in a CSG tree is always a valid solid A valid solid implies a solid of whichinside and outside region can be clearly identified An example of an invalid solid would be asolid with a strut edge In this case the notion of inside and outside is not clear around thevertex where the strut edge is attached1048698 CSG representation of a solid can be always converted to the corresponding B-rep Thus CSGtree representation can be interfaced to the application programs written for B-Rep1048698 Parametric modeling can be realized very easily by changing the parameter values of theassociated primitives

DisadvantageHowever the CSG tree structure has the following disadvantages1048698 Since CSG tree structure stores the history of applying the Boolean operations only theBoolean operations are allowed in the modeling process With Boolean operations alone therange of shapes to be modeled is restricted severely Furthermore the convenient localmodification functions such as lifting and rounding can not be used1048698 It requires a large amount of computation to derive the information on the boundary surfacestheir boundary edges and the connectivity between these boundary entities from CSG treerepresentation Unfortunately there are many applications which require these boundaryinformation One example would be a display of solids Whether a shaded image or a linedrawing of a solid is to be displayed as explained in Chapter 3 the information on the facesandor edges is required Thus CSG tree representation has been assumed to be inappropriatefor the interactive display and manipulation of solids Another example would be thecalculation of NC tool paths to machine the surfaces of a solid by a milling machine In thisapplication the information on the surface to be machined and its boundary edges isnecessary Furthermore the adjacent surfaces need to be derived for the gouging detection Itis not a simple task to derive all these boundary information from a CSG tree representation ofa solid

4(1)During building the Oct-tree representation of a solid when deciding the color of an octant (ie whether it is inside outside or intersecting the boundary of the object) you can not simply decide it by merely looking at the center point of the octant why (2)Then how do you decide the color of the octant Whatrsquos the major disadvantage of it

Answer (1)Because only decide the center point the area that inside or outside can not be known clearly This cause large error To decide the color of the Octant in the following way Step 1Make a hexahedron H that completely encloses the solidStep 2Divide H into 8 octantsStep 3For each octant obtained in step 2 mark it by one of three colorsWhite completely outside the solidBlack completely inside the solidGray partially inside the solidStep 4For each gray octant obtained in step 3 go to step 2 until either no gray octants are left or the size of the octant reaches a minimum threshold(2)The major disadvantage is time consuming

5 Euler-Poincare formula of solids with voids (1) What is the relationship between V E F H and P of the solid below (The small cube is a hollow part also called a void)(2) Let W be the number of voids in the solid establish the relationship between V E F H P and W (Hint first exam the case of two voids and compare with that of a single void)

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids

2 Give at least two types of primitives in solid modeling that can also be defined as sweeps

Answer CylinderTorus

3Suppose a solid is stored in computer with CSG data structure When it is displayed on the screen computer needs its faces to triangulate them and render those triangles on the screen Does CSG have those faces Why If not how does the computer get those faces What is the main disadvantage of this What is the main advantage of CSG

Answer CSG does not have those faces because it only stores the history of applying Boolean operations on the primitives It is necessary to transform the CSG into other types of representation

Answer

Advantage The CSG tree data structure has the following advantages1048698 Data structure is simple and stores compact data Accordingly the management of data iseasy1048698 The solid stored in a CSG tree is always a valid solid A valid solid implies a solid of whichinside and outside region can be clearly identified An example of an invalid solid would be asolid with a strut edge In this case the notion of inside and outside is not clear around thevertex where the strut edge is attached1048698 CSG representation of a solid can be always converted to the corresponding B-rep Thus CSGtree representation can be interfaced to the application programs written for B-Rep1048698 Parametric modeling can be realized very easily by changing the parameter values of theassociated primitives

DisadvantageHowever the CSG tree structure has the following disadvantages1048698 Since CSG tree structure stores the history of applying the Boolean operations only theBoolean operations are allowed in the modeling process With Boolean operations alone therange of shapes to be modeled is restricted severely Furthermore the convenient localmodification functions such as lifting and rounding can not be used1048698 It requires a large amount of computation to derive the information on the boundary surfacestheir boundary edges and the connectivity between these boundary entities from CSG treerepresentation Unfortunately there are many applications which require these boundaryinformation One example would be a display of solids Whether a shaded image or a linedrawing of a solid is to be displayed as explained in Chapter 3 the information on the facesandor edges is required Thus CSG tree representation has been assumed to be inappropriatefor the interactive display and manipulation of solids Another example would be thecalculation of NC tool paths to machine the surfaces of a solid by a milling machine In thisapplication the information on the surface to be machined and its boundary edges isnecessary Furthermore the adjacent surfaces need to be derived for the gouging detection Itis not a simple task to derive all these boundary information from a CSG tree representation ofa solid

4(1)During building the Oct-tree representation of a solid when deciding the color of an octant (ie whether it is inside outside or intersecting the boundary of the object) you can not simply decide it by merely looking at the center point of the octant why (2)Then how do you decide the color of the octant Whatrsquos the major disadvantage of it

Answer (1)Because only decide the center point the area that inside or outside can not be known clearly This cause large error To decide the color of the Octant in the following way Step 1Make a hexahedron H that completely encloses the solidStep 2Divide H into 8 octantsStep 3For each octant obtained in step 2 mark it by one of three colorsWhite completely outside the solidBlack completely inside the solidGray partially inside the solidStep 4For each gray octant obtained in step 3 go to step 2 until either no gray octants are left or the size of the octant reaches a minimum threshold(2)The major disadvantage is time consuming

5 Euler-Poincare formula of solids with voids (1) What is the relationship between V E F H and P of the solid below (The small cube is a hollow part also called a void)(2) Let W be the number of voids in the solid establish the relationship between V E F H P and W (Hint first exam the case of two voids and compare with that of a single void)

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids

3Suppose a solid is stored in computer with CSG data structure When it is displayed on the screen computer needs its faces to triangulate them and render those triangles on the screen Does CSG have those faces Why If not how does the computer get those faces What is the main disadvantage of this What is the main advantage of CSG

Answer CSG does not have those faces because it only stores the history of applying Boolean operations on the primitives It is necessary to transform the CSG into other types of representation

Answer

Advantage The CSG tree data structure has the following advantages1048698 Data structure is simple and stores compact data Accordingly the management of data iseasy1048698 The solid stored in a CSG tree is always a valid solid A valid solid implies a solid of whichinside and outside region can be clearly identified An example of an invalid solid would be asolid with a strut edge In this case the notion of inside and outside is not clear around thevertex where the strut edge is attached1048698 CSG representation of a solid can be always converted to the corresponding B-rep Thus CSGtree representation can be interfaced to the application programs written for B-Rep1048698 Parametric modeling can be realized very easily by changing the parameter values of theassociated primitives

DisadvantageHowever the CSG tree structure has the following disadvantages1048698 Since CSG tree structure stores the history of applying the Boolean operations only theBoolean operations are allowed in the modeling process With Boolean operations alone therange of shapes to be modeled is restricted severely Furthermore the convenient localmodification functions such as lifting and rounding can not be used1048698 It requires a large amount of computation to derive the information on the boundary surfacestheir boundary edges and the connectivity between these boundary entities from CSG treerepresentation Unfortunately there are many applications which require these boundaryinformation One example would be a display of solids Whether a shaded image or a linedrawing of a solid is to be displayed as explained in Chapter 3 the information on the facesandor edges is required Thus CSG tree representation has been assumed to be inappropriatefor the interactive display and manipulation of solids Another example would be thecalculation of NC tool paths to machine the surfaces of a solid by a milling machine In thisapplication the information on the surface to be machined and its boundary edges isnecessary Furthermore the adjacent surfaces need to be derived for the gouging detection Itis not a simple task to derive all these boundary information from a CSG tree representation ofa solid

4(1)During building the Oct-tree representation of a solid when deciding the color of an octant (ie whether it is inside outside or intersecting the boundary of the object) you can not simply decide it by merely looking at the center point of the octant why (2)Then how do you decide the color of the octant Whatrsquos the major disadvantage of it

Answer (1)Because only decide the center point the area that inside or outside can not be known clearly This cause large error To decide the color of the Octant in the following way Step 1Make a hexahedron H that completely encloses the solidStep 2Divide H into 8 octantsStep 3For each octant obtained in step 2 mark it by one of three colorsWhite completely outside the solidBlack completely inside the solidGray partially inside the solidStep 4For each gray octant obtained in step 3 go to step 2 until either no gray octants are left or the size of the octant reaches a minimum threshold(2)The major disadvantage is time consuming

5 Euler-Poincare formula of solids with voids (1) What is the relationship between V E F H and P of the solid below (The small cube is a hollow part also called a void)(2) Let W be the number of voids in the solid establish the relationship between V E F H P and W (Hint first exam the case of two voids and compare with that of a single void)

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids

Answer

Advantage The CSG tree data structure has the following advantages1048698 Data structure is simple and stores compact data Accordingly the management of data iseasy1048698 The solid stored in a CSG tree is always a valid solid A valid solid implies a solid of whichinside and outside region can be clearly identified An example of an invalid solid would be asolid with a strut edge In this case the notion of inside and outside is not clear around thevertex where the strut edge is attached1048698 CSG representation of a solid can be always converted to the corresponding B-rep Thus CSGtree representation can be interfaced to the application programs written for B-Rep1048698 Parametric modeling can be realized very easily by changing the parameter values of theassociated primitives

DisadvantageHowever the CSG tree structure has the following disadvantages1048698 Since CSG tree structure stores the history of applying the Boolean operations only theBoolean operations are allowed in the modeling process With Boolean operations alone therange of shapes to be modeled is restricted severely Furthermore the convenient localmodification functions such as lifting and rounding can not be used1048698 It requires a large amount of computation to derive the information on the boundary surfacestheir boundary edges and the connectivity between these boundary entities from CSG treerepresentation Unfortunately there are many applications which require these boundaryinformation One example would be a display of solids Whether a shaded image or a linedrawing of a solid is to be displayed as explained in Chapter 3 the information on the facesandor edges is required Thus CSG tree representation has been assumed to be inappropriatefor the interactive display and manipulation of solids Another example would be thecalculation of NC tool paths to machine the surfaces of a solid by a milling machine In thisapplication the information on the surface to be machined and its boundary edges isnecessary Furthermore the adjacent surfaces need to be derived for the gouging detection Itis not a simple task to derive all these boundary information from a CSG tree representation ofa solid

4(1)During building the Oct-tree representation of a solid when deciding the color of an octant (ie whether it is inside outside or intersecting the boundary of the object) you can not simply decide it by merely looking at the center point of the octant why (2)Then how do you decide the color of the octant Whatrsquos the major disadvantage of it

Answer (1)Because only decide the center point the area that inside or outside can not be known clearly This cause large error To decide the color of the Octant in the following way Step 1Make a hexahedron H that completely encloses the solidStep 2Divide H into 8 octantsStep 3For each octant obtained in step 2 mark it by one of three colorsWhite completely outside the solidBlack completely inside the solidGray partially inside the solidStep 4For each gray octant obtained in step 3 go to step 2 until either no gray octants are left or the size of the octant reaches a minimum threshold(2)The major disadvantage is time consuming

5 Euler-Poincare formula of solids with voids (1) What is the relationship between V E F H and P of the solid below (The small cube is a hollow part also called a void)(2) Let W be the number of voids in the solid establish the relationship between V E F H P and W (Hint first exam the case of two voids and compare with that of a single void)

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids

DisadvantageHowever the CSG tree structure has the following disadvantages1048698 Since CSG tree structure stores the history of applying the Boolean operations only theBoolean operations are allowed in the modeling process With Boolean operations alone therange of shapes to be modeled is restricted severely Furthermore the convenient localmodification functions such as lifting and rounding can not be used1048698 It requires a large amount of computation to derive the information on the boundary surfacestheir boundary edges and the connectivity between these boundary entities from CSG treerepresentation Unfortunately there are many applications which require these boundaryinformation One example would be a display of solids Whether a shaded image or a linedrawing of a solid is to be displayed as explained in Chapter 3 the information on the facesandor edges is required Thus CSG tree representation has been assumed to be inappropriatefor the interactive display and manipulation of solids Another example would be thecalculation of NC tool paths to machine the surfaces of a solid by a milling machine In thisapplication the information on the surface to be machined and its boundary edges isnecessary Furthermore the adjacent surfaces need to be derived for the gouging detection Itis not a simple task to derive all these boundary information from a CSG tree representation ofa solid

4(1)During building the Oct-tree representation of a solid when deciding the color of an octant (ie whether it is inside outside or intersecting the boundary of the object) you can not simply decide it by merely looking at the center point of the octant why (2)Then how do you decide the color of the octant Whatrsquos the major disadvantage of it

Answer (1)Because only decide the center point the area that inside or outside can not be known clearly This cause large error To decide the color of the Octant in the following way Step 1Make a hexahedron H that completely encloses the solidStep 2Divide H into 8 octantsStep 3For each octant obtained in step 2 mark it by one of three colorsWhite completely outside the solidBlack completely inside the solidGray partially inside the solidStep 4For each gray octant obtained in step 3 go to step 2 until either no gray octants are left or the size of the octant reaches a minimum threshold(2)The major disadvantage is time consuming

5 Euler-Poincare formula of solids with voids (1) What is the relationship between V E F H and P of the solid below (The small cube is a hollow part also called a void)(2) Let W be the number of voids in the solid establish the relationship between V E F H P and W (Hint first exam the case of two voids and compare with that of a single void)

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids

4(1)During building the Oct-tree representation of a solid when deciding the color of an octant (ie whether it is inside outside or intersecting the boundary of the object) you can not simply decide it by merely looking at the center point of the octant why (2)Then how do you decide the color of the octant Whatrsquos the major disadvantage of it

Answer (1)Because only decide the center point the area that inside or outside can not be known clearly This cause large error To decide the color of the Octant in the following way Step 1Make a hexahedron H that completely encloses the solidStep 2Divide H into 8 octantsStep 3For each octant obtained in step 2 mark it by one of three colorsWhite completely outside the solidBlack completely inside the solidGray partially inside the solidStep 4For each gray octant obtained in step 3 go to step 2 until either no gray octants are left or the size of the octant reaches a minimum threshold(2)The major disadvantage is time consuming

5 Euler-Poincare formula of solids with voids (1) What is the relationship between V E F H and P of the solid below (The small cube is a hollow part also called a void)(2) Let W be the number of voids in the solid establish the relationship between V E F H P and W (Hint first exam the case of two voids and compare with that of a single void)

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids

5 Euler-Poincare formula of solids with voids (1) What is the relationship between V E F H and P of the solid below (The small cube is a hollow part also called a void)(2) Let W be the number of voids in the solid establish the relationship between V E F H P and W (Hint first exam the case of two voids and compare with that of a single void)

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids

Answer V =24 E = 36 F = 16 H = 2 P = 1 By Euler-Poincare formula that is V-E+F-H =2(1-P)However 24-36+16-2= 2 not equal to 2(1-1)Therefore this situation does not satisfy the formula

we add an unknown x to balance the formula that is

V-E+F-H = 2(1-P)+x here x=2 for one void Assume there are two voids in the solid V=32 E=48

F=22 H=2 P=1in to above formula x=4 Similarly for solid with 3 x=6 solid with 4 x=8We could get the equationV-E+F-H = 2(1-P)+2WW is the number of the voids