# Challenges of Engineering Applications of Descriptive Geometry

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

_{1}, K

_{2}} consisting of mutually perpendicular image planes together with the first projector line v

_{1}perpendicular to the first image plane, and the second projector line v

_{2}perpendicular to the second image plane defines a Monge projection.

## 2. Method of Ensuring the Bijectivity of Monge’s Representation by Defining the Directions of the Views

#### 2.1. Examining the Spatial Curve

_{1}and K

_{2}of the Monge representation, respectively. In this arrangement, the y coordinate axis will be the intersection straight line of the two image planes, namely the x

_{12}axis.

**Theorem 1.**

_{1}(y) and y→f

_{2}(y), respectively, in the corresponding Cartesian coordinate planes of the image planes, where x = f

_{1}(y) and z = f

_{2}(y) i.e., its points have coordinates P(f

_{1}(y), y, f

_{2}(y)), then any part of the curve g can be clearly reconstructed from its images.

**Proof of Theorem 1.**

_{1}(y) ≡ P′ fitting to the plane [xy] ≡ K

_{1}and one f

_{2}(y) ≡ P″ fitting to the plane [yz] ≡ K

_{2}. These lines assigning P′ and P″ to a single y value is located perpendicular to the coordinate axis y ≡ x

_{12}axis. Thus, after the merging of the image planes K

_{1}and K

_{2}, the (P′, P″) forms an ordered pair of points to which only one spatial point P belongs. Therefore, any point of the curve g and thus any part of it can be clearly reconstructed. □

**Corollary 1.**

_{1}(y) and y→f

_{2}(y), then no single profile plane of the Monge projection intersects more than one point each from g′ and g″ each.

**Theorem 2.**

_{1}(y) and z = f

_{2}(y), respectively the assignments and y→f

_{1}(y) and y→f

_{2}(y) are not functions, then there is a part of curve g, that cannot be clearly reconstructed from its two images.

**Proof of Theorem**

**2.**

_{1}(y) ≡ P′ points lying on the plane [xy] ≡ K

_{1}and several points f

_{2}(y) ≡ P″ lying on the plane [yz] ≡ K

_{2}. These image points P′ and P″ fitting to a recall line perpendicular to the axis y ≡ x

_{12}can optionally be arranged to form ordered pairs of points, to which the spatial points P belong. So, there exists a point of the curve g and a neighbor of this point that cannot be reconstructed from only two images. □

**Corollary 2.**

_{1}(y) and z = f

_{2}(y) then

- 1.
- there exist profile planes of the Monge projection that intersect the curve g in more than one point, and
- 2.
- there is at least one profile plane tangent to g.

**Theorem 3.**

_{1}(y) or z = f

_{2}(y) but the other image is a double projection, and this can be written as functions z = f

_{2}(y) or x = f

_{1}(y), then any part of g can be clearly reconstructed from only two images.

**Proof of Theorem 3.**

- Assume that from the image curves g′ and g″ of the curve g, g′ is a double projection, and it can be written as a function x = f
_{1}(y), and the g″ cannot be written as a function z = f_{2}(y) on the corresponding Cartesian coordinate plane. In this case, a single y corresponds to a point f_{1}(y) ≡ P′ fitting to the plane [xy] ≡ K_{1}and several points f_{2}(y) ≡ P″ fitting to the plane [yz] ≡ K_{2}. These one P′ and several P″ points located perpendicular to the y ≡ x_{12}axis can be assigned to each other to form ordered pairs of points, to which several spatial P points belong. Thus, any point of the curve g and thus any part of it can be clearly reconstructed from only two images. - If the image curve g″ is a double projection which can be written as a function z = f
_{2}(y) and g′ cannot be written as a function x = f_{1}(y), the proof is the indices 1 and 2, as well as ′, and ″ can be done in the same way as in case 1 by exchanging the signs. □

**Corollary 3.**

_{1}(y) or z = f

_{2}(y), but the other image is a double projection, and this can be written as a function z = f

_{2}(y) or x = f

_{1}(y), then there is at least one profile plane that touches g.

**Theorem 4.**

**Proof of Theorem 4.**

_{0}with parameter u

_{0}of its images is not in the direction of the recall line, namely the recall line intersects the examined curve in point P

_{0}. So, there is no point P

_{0}of the curve g with parameter u

_{0}, in the region of which all points P

_{−1}and P

_{1}with parameters u

_{−1}and u

_{1}are located on one side of the recall line of the point P

_{0}, where u

_{−1}< u

_{0}< u

_{1}. This means that there is no curve segment that has two points on a recall straight line. Because each recall straight line has only one point on the curve g, any point on curve g can be reconstructed from its two images, which means that the representation of the curve is bijective. □

#### 2.2. Correspondence between Ordered Orthogonal Projections and Real Number Triplets

**Theorem 5.**

**Proof of Theorem 5.**

_{12}axis, namely the intersection line of the two image planes can be characterized by 2 free parameters, for example two spherical coordinates, the image planes can be described by 1 free parameter in the possibilities of rotation around the x

_{12}axis. Consequently, Monge projections can be described with 3 free parameters in addition to the previous restrictions. After all this, a number triple has been assigned to each Monge projection, with its elements having the geometric meaning of an angle. However, before assigning these, it is necessary to define the directed angles of the straight line.

#### 2.2.1. Directed Angles of the Straight Line

**Definition 2.**

^{+}can be rotated towards its first projection e′ of the straight line e on the plane [xy] in the direction of the semi-axis y

^{+}, counter- clockwise as viewed from a point at infinity on the semi-axis z

^{+}(Figure 6). The first directed angle should be α = 0, if the straight line e coincides with the axis z. The first directed angle of the straight line bypassing the origin point O is the same as that of the one running parallel to it and passing through the origin point O.

**Definition 3.**

^{+}can be rotated to its second projection e″ of the straight line e on the plane [yz] in the direction of the semi-axis z

^{+}, counter- clockwise viewed from a point at infinity on the semi-axis x

^{+}(Figure 6). The second directed angle should be β = 0, if the straight line e coincides with the axis x. The second directed angle of the straight line e bypassing the origin point O is the same as that of the one running parallel to it and passing through the origin O.

**Definition 4.**

^{+}can be rotated to the third projection e‴ of the straight line e on the plane [zx] in the direction of the semi-axis x

^{+,}counter-clockwise viewed from the infinity point of the semi-axis y

^{+}(Figure 6). The third directed angle should be interpreted according to γ = 0, if the straight line e coincides with the axis y. The third directed angle of the straight line e bypassing the origin point O is the same as the third directed angle of the straight line parallel to it passing through the origin point O.

**Theorem 6.**

_{1}, K

_{2}} of a Monge projection fit to a fix point O of the space, then the Monge projection is defined by its projector lines v

_{1}and v

_{2}passing through the origin O.

**Proof of Theorem 6.**

_{1}and K

_{2}will be perpendicular v

_{1}, and v

_{2,}respectively. There are an infinite number of these pairs of planes, but the Monge systems are derived from each other with a parallel display, namely these are equivalent from the point of view of the present examination. □

**Theorem 7.**

_{1}, K

_{2}and projector lines v

_{1}, v

_{2}is also bijective or nonbijective with respect to the given curve g.

**Proof of Theorem 7.**

_{1}and K

_{2}and the projector lines v

_{1}, v

_{2}the image curves g′ and g″ of the g curve do not modify, only their comma notations are exchanged, namely g′ becomes g″ and g″ becomes g′ because of the symmetry property. □

#### 2.2.2. The Relationship between the Triplets of Directed Angles and the Monge Projections

**Theorem 8.**

_{1}and v

_{2}fulfil both the v

_{1}∊ [zx] and v

_{2}∉ [zx] conditions, define three independent parameters (α, β, γ) in a space fixed Cartesian coordinate system O[x, y, z] as follows: the first directed angle of the first projector line v

_{1}of the Monge projection should be parameter α, while the second directed angle should be parameter β, and the third directed angle of the second projector line v

_{2}should be parameter γ.

**Remark 1.**

**Proof of Theorem 8.**

_{1}∉ [zx], and 2, when v

_{1},v

_{2}∈ [zx].

- The first projector line v
_{1}is not on the coordinate plane [zx] of the O[x, y, z] Cartesian coordinate system. The rotation of x^{+}on the plane [xy] by α into the direction of y^{+}yields the first image v_{1}′ of the projector line v_{1}, to which the plane V_{1}fits and is perpendicular to the plane [xy]. Then, the rotation of y^{+}on the plane [yz] by into the direction of z^{+}results the second image v_{1}″ of the projector line v_{1}, to which the plane V_{2}fits and is perpendicular to the plane [yz]. Since the assumption v_{1}∉ [zx] is hold, there exists a straight line of intersection of planes V_{1}and V_{2}, serving as the first projector line v_{1}of the sought Monge projection. Then, the rotation of z^{+}on the plane [zx] with γ into the direction of x^{+}yields the third image v_{2}‴ of the projector line v_{2}, to which the plane V_{3}fits and is perpendicular to the plane [zx]. Again, due to our assumption v_{1}∉ [zx], the plane N perpendicular to v_{1}can never coincides with plane V_{3}, so there is a straight line of intersection of planes N and V_{3}. This is the second projector line v_{2}of the Monge projection (Figure 7).Based on Theorem 6, the projector lines v_{1}and v_{2}determine the Monge projection. - The first and second projector lines v
_{1}and v_{2}fit on the coordinate plane [zx] of the Cartesian coordinate system O[x, y, z]. In this case, the corresponding Monge projection is derived from the directed angles triplet (α, β, γ) by rotating z^{+}in the direction of x^{+}by γ on the plane [zx]. This will be the second projector line v_{2}of the sought Monge projection, and then the first projector line of v_{1}is perpendicular to it as shown in Figure 8. □

_{1}and v

_{2}determines the Monge projection. Since v

_{1}has a first and a second directed angle and v

_{2}has a third directed angle by definition, each Monge projection can be assigned one number triple only. According to the reverse assignment, each point of the Monge cuboid defines a triplet of real numbers, the geometric meaning of which is a directed angle, which provides a Monge projection respected to the given curve.

**Definition 5.**

#### 2.3. Application of the Method

- The curve must be positioned in a fixed O[x, y, z] initial Cartesian coordinate system.
- The direction cone formed from the directions of the tangents of the curve must be determined, that is, the tangents of the curve must be moved parallel to themselves into a properly selected point of the z axis, such as the origin O, by parallel displacement.
- Examining the mutual positions of the profile planes of the Monge projections and the directional cone, it is necessary to find the cases when they do not have a common line.

#### 2.3.1. Procedure for Representing a Straight Line

- In the first case, the profile plane labeled P
_{1}fits on the [zx] coordinate plane, namely P_{1}≡ [zx], as well as the projector lines v_{1}and v_{2}do not fit on either the z or x coordinate axes. In this case, the subset to be found is declared by angle triples satisfying the following conditions

- 2.
- In the second case, the profile plane labeled P2 fits on the [yz] coordinate plane, namely P2 ≡ [yz], as well as the projector lines v1 and v2 do not fit on either the y or x coordinate axes. In this case, the subset to be found is declared by angle triples satisfying the following conditions

- 3.
- In the third case, as outlined in Figure 10, the profile plane in position P
_{3}contains on the z coordinate axis, but does not contain either of the x or y coordinate axes, and none of the projector lines lie onto the z coordinate axis.

_{3}the direction vector of the first projector line v

_{1}should be

**v**

_{1}(v

_{1x}, v

_{1y,}v

_{1z}), the direction vector of the second projection line v

_{2}should be

**v**

_{2}(v

_{2x}, v

_{2y,}v

_{2z}) as shown in Figure 11.

_{3}, the

**n**normal vector of the first projector plane V

_{1}lying to the v

_{1}and v

_{2}projector lines should be

**n**= (v

_{1y}, −v

_{1x}, 0).

**v**

_{2}⊥

**v**

_{1}and

**v**

_{2}⊥

**n**, due to the following relation

**v**

_{2}second projector line are

#### 2.3.2. Procedure for Representing a Circle

_{1}or the second projector line v

_{2}lies in the plane [xy], then one of the images of the circle is a diameter-length section, namely a double projection, so it can be described as a function on the corresponding Cartesian coordinate plane, while the other image is a circle or an ellipse. In this exceptional case, the representation of the circle with the given location is bijective. For the reasons listed earlier, there are two cases to be considered:

- 2.
- Where v
_{2}∈ [xy] and v_{1}∉ [xy], the fulfillment of the v_{1}∈z condition is determined by the directed angles α = 0, β = π/2 and γ = π/2 as shown in Figure 13a), so the circle can be clearly represented. If v_{2}∈ [xy] and v_{1}∉ [xy] are fulfilled as shown in Figure 13b), the representation of the given circle is bijective.

#### 2.3.3. Procedure for Representing a Helix

^{+}, 0 ≤ φ < 2π.

**r**

_{e}

_{2}contains two generatrixes of the direction cone of the tangents as shown in Figure 15, the representation of a pitch of the helix results two tangents in profile direction.

_{2}.

_{1}in Figure 15, then at the point belonging to the profile-oriented tangent straight line of the helix, the tangent straight line intersects the image curves, namely this point is the singular points of the image curves, due to the cyclicity of the helix and its images. In this case, any part of the helix can be clearly reconstructed from its two images. If the cone of the tangential directions of the helix does not have a single common straight line with the profile plane of the Monge projection, namely the profile plane in position P

_{0}as shown in Figure 15. In this case, the images of the helix are curtate cycloids (Figure 16) or planar curves, which are in affine relationship to the curtate cycloids and contain inflection points. In the case of such a Monge projection, any segment of the helix can be unambiguously reproduced from its first and second images.

**n**(n

_{x}, n

_{y}, n

_{z}) are perpendicular to the tangent planes of the cone of tangent directions of the helix. The normal vectors placed at the origin O create the normal cone, as shown in Figure 17.

**n**. Let us have the vector

**n**(n

_{x}, n

_{y}, n

_{z}) as the unit vector, namely |

**n**| = 1, and

**z**(0, 0, 1) defined as one coinciding with the axis z.

**n**(n

_{x}, n

_{y}, sinω) normal vectors satisfying relations (22) and (23), and to provide the coordinates α, β, γ of the points of the Monge cuboid that define bijective Monge projections for the given helix. Profile planes perpendicularly to the normal vectors, then the Monge projections belonging to the profile planes, and finally to specify some conditions of the relations between the α, β, γ coordinates of the points of the Monge cuboid.

_{1}and v

_{2}are perpendicular to each other, so their direction vectors

**v**

_{1}and

**v**

_{2}are also perpendicular to each other, as a result of which the following relation is fulfilled

**v**

_{1}(v

_{1x}, v

_{1y}, v

_{1z}) and

**v**

_{2}(v

_{2x}, v

_{2y}, v

_{2z})

- I.
- In the first part of the examination, the assumptions α,β,γ ≠ 0,π/2,π are considered.

_{z}= sinω ≠ 0 is fulfilled. In this case, the substitution with the equations (9) and (28)–(33) gives

_{z}= sinω gives

- II.
- In the second part of the examination, the relation (38) should be modified for the coordinates n
_{x}, n_{y}, n_{z}of the vectors**n**belonging to the previously excluded cases of α, β, γ ≠ 0, π/2, π;

- 1.(i).
- The points determined by the coordinates

- α = π, β = 0, γ = π belong to the nonbijective subset of the Monge cuboid;
- α = 0, β = π/2, γ = π/2, also belong to the nonbijective subset of the Monge cuboid;

- 1.(ii).
- Among the points defined by coordinates corresponding to the conditions 0 < α < π, β = π, 0 < γ ≤ π, those whose coordinates correspond to the following sub-criteria, such as the

- 0 < α < π, β = π, γ = π/2, always result a bijective Monge projection due to the condition 0 < ω<π;
- 0 < α < π, β = π, γ = π, always result a nonbijective Monge projections to the given helix, because of the circle shown second image;
- 0 < α < π, β = π, 0 < γ < π/2, π/2 < γ < π, should be assumed in these cases, so that |
**v**_{1}| = 1.

**v**

_{1}and

**v**is

_{2}- 1.(iii).
- In the case of points defined by the coordinates corresponding to the conditions 0 < α < π, 0 < β < π/2, π/2 < β < π, γ = π, those whose coordinates correspond to the following sub-criteria

- α = π/2, 0 < β < π/2, π/2 < β < π, γ = π always results nonbijective Monge projections;
- 0 < α < π/2, π/2 < α < π, 0 < β < π/2, π/2 < β < π, γ = π, the v
_{2x}= 0, and n_{x}, n_{z}, v_{1x}, v_{1z}≠ 0,

_{z}= sinω. Then, based on (8), (9), (26), (29), (31), (33) the coordinates of the normal vector will be as follows

- 2.(i).
- Among the points defined by coordinates corresponding to the conditions α = π/2, 0 < β < π/2 and π/2 < β < π, 0 < γ< π, those whose coordinates correspond to the following sub-criteria:

- in the case of α = π/2, 0 < β < π/2, π/2 < β < π and γ = π/2, the v
_{1x}= 0, and the v_{2}∈ x, so v_{2y}, v_{2z}= 0

_{2x}= 1. Furthermore, if the identities n

_{x}= 0, n

_{y}= v

_{1z}and n

_{z}= −v

_{1y}= sinω, plus n

_{y}= −sinω·tgβ are fulfilled, every triplet that satisfies

- In the case of α = π/2, 0 < β < π/2, π/2 < β < π and 0 < γ < π/2, π/2 < γ < π, the v
_{1x}= 0. Since γ ≠ π/2, therefore v_{2}∉ [xy], so n_{x}≠0, and since γ ≠ 0, π and the v_{2}∉ [yz], consequently n_{z}≠ 0.

- 2.(ii).
- The points with coordinates corresponding to the conditions 0 < α < π/2, π/2 < α < π, 0 < β < π/2, π/2 < β < π, γ = π/2 define Monge projections, the second projector line v
_{2}of which lies on the [xy] plane, so v_{2z}= 0. In the case of triplets also satisfying the condition

#### 2.3.4. Procedure for Representing a Cubic Curve

**p**

_{0}and

**p**

_{3}pointing to the starting and ending points of the curve P

_{0}and P

_{3}be given, as well as the corresponding starting and ending tangent vectors

**t**

_{0}and

**t**

_{1}, as shown in Figure 23.

**n**(n

_{x}, n

_{y}, n

_{z}) of the planes containing any of the tangent vectors are perpendicular to the tangent vectors lying in it; therefore, the following equation is fulfilled, namely

**n**(n

_{x}, n

_{y}, n

_{z}), for which the quadratic equation for the parameter u has no solution.

**n**(n

_{x}, n

_{y}, n

_{z}) in Equation (57) must be determined, for which the quadratic equation with respect to the u parameter has no solution in the case of the specified e

_{ij}(i = 1,2,3 and j = x,y,z) values. The profile planes determined by such normal vectors do not have a tangent to the examined curve, therefore in the Monge projections related to these profile planes the curve representation is bijective. For the normal vector of the profile plane of all bijective Monge projections, the value of the discriminant of equality (57) is negative, that is

_{i}and c

_{ij}(i,j = 1,2,3) giving a new form of the inequality:

## 3. Result and Application in Mechanical Engineering Practice

_{3}and P

_{0}on the addendum and root cylinder, so that the points P

_{2}and P

_{1}between them could be defined proportionally to their distance from the axis. The interpolating Bezier curve has four selected points with their position vectors

**p**

_{0},

**p**

_{1,}

**p**

_{2},

**p**

_{3}on the cutting edge of the hob, and its parameters u

_{0}, u

_{1}, u

_{2}, u

_{3}satisfy the condition u

_{i}≠ u

_{j}, if i ≠ j, as well as u

_{0}= 0 and u

_{3}= 1. The coordinates of the position vectors

**b**

_{0},

**b**

_{1,}

**b**

_{2},

**b**

_{3}of the control points B

_{0}, B

_{1,}B

_{2}, B

_{3}have to be calculated, which will determine the interpolation Bezier curve passing through the selected points so that the following equation is fulfilled

_{0}, P

_{1}, P

_{2}and P

_{3}, it can be defined by the following equation

**b**

_{i}vectors (i = 0,…,3) pointing to control points B

_{0}, B

_{1,}B

_{2}, B

_{3}of the Bezier curve passing through the selected points P

_{0}, P

_{1,}P

_{2}, P

_{3}can be calculated based on Equation (65). The relationship between the

**p**

_{0},

**p**

_{1,}

**p**

_{2},

**p**

_{3}position vectors of the control points of the Bezier curve and the position vectors of the start and end points

**p**

_{0}and

**p**

_{3}, as well as the start and end tangents

**t**

_{0}and

**t**

_{3}of the Hermite arc can be characterized by the following relations based on the literature [49] and according to the guidance of Figure 27.

**n**

_{h}of the face surface is determined from the coordinates of the points P

_{3}and P

_{0}measured on the cutting edge, as well as the coordinates of the points L

_{3}and L

_{0}measured on the root cylinder curve by the cross product of the difference vectors of the position vectors pointing to the points according to the following relation

**n**

_{h}normal vector and the

**v**

_{1}and

**v**

_{2}direction vectors of the projector lines can be determined as follows

_{1}and v

_{2}projecting into the direction of the axis, can have a maximum value ω

_{1}as shown in Figure 26b). This means that the minimum value of the angles ε

_{h1}and ε

_{h2}between the direction vectors of the projector lines and the normal vector

**n**

_{h}must be ${90}^{\mathrm{o}}-{\mathsf{\omega}}_{1}$ for the chip groove angle of size ω

_{1}, as it can be written as

## 4. Discussion

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic layout of the relationship between the images taken by CCD cameras, arranged perpendicularly to each other and facing the v

_{1}and v

_{2}projector lines, and Monge mapping.

**Figure 6.**The first directed angle α, the second directed angle β and the third directed angle γ of the straight line e.

**Figure 7.**The relationship between the triplets of angle-parameters (α, β, γ) and the projector lines v

_{1}, v

_{2}of Monge projections in a Cartesian coordinate system O[x, y, z] fixed in space.

**Figure 8.**The lines v

_{1}, v

_{2}of Monge projections fitting to the plane [zx] of the space fixed Cartesian coordinate system O[x, y, z].

**Figure 10.**The straight line e coincides with axis z, and the profile planes fitted to it are in positions P

_{1}, P

_{2}and P

_{3}with contained projector lines v

_{i}and v

_{j}(i,j = 1,2 and i ≠ j), which do not coincide with any of the coordinate axes.

**Figure 11.**The relationship between the triple of angle parameters (α, β, γ) and the projector lines v

_{1}, v

_{2}of Monge projections in the space fixed Cartesian coordinate system O[x, y, z].

**Figure 12.**The circle contained by the plane [xy] with its center origin. (

**a**) The projector lines v

_{1}and v

_{2}coincide with the coordinate axes x and z, respectively. (

**b**) The projector line v

_{1}is contained by the coordinate plane [xy] but not coincides with any of the coordinate axes and the projector line v

_{2}is on the normal plane N of the v

_{1}projector line.

**Figure 13.**The circle contained by the coordinate plane [xy] with its center in the origin O and (

**a**) the v

_{1}projector line coincide with the coordinate axis z, respectively the v

_{2}projector line coincide with the coordinate axis x. (

**b**) the v

_{1}projector line is contained on the coordinate plane [xy] but not coincide with any of the coordinate axes and the projector line v

_{2}is on its normal plane N.

**Figure 15.**The schematic illustration of the relative positions between the direction cone of the tangents with the z-axis helix and the profile planes of the Monge projections, when they have two common component lines in the P

_{2}position, one common component line in the P

_{1}position and no common component line in the case of the P

_{0}position of the profile plane.

**Figure 16.**A complete turn of a curtate cycloid created by the inner point C

_{0}of the circle rolling on a straight line without slipping.

**Figure 18.**Demonstration of the bijective representation of the helix when of ω = π/4 in the Monge projection assigned to the triplet of directed angles (π/6, π/6, π) by the computer program developed for this purpose.

**Figure 19.**Demonstration of the bijective representation of the helix when ω = π/4 in the Monge projection assigned to the triplet of directed angles (π/4, π/4, π/4) by using the computer program created for this procedure.

**Figure 20.**A nonbijective representation of the helix in that Monge projection, which is determined by its projector lines directions calculated from the three directed angles (π/3, π/4, 2π/3) with the computer program developed for this purpose, in the case of ω = π/4.

**Figure 21.**A nonbijective representation of the helix in that Monge projection, which is determined by its projector lines directions calculated from the three directed angles (π/3, π/3, π/2) with the computer program developed for this purpose, in the case of ω = π/4.

**Figure 22.**Inner points of the bijective subset of the Monge cuboid are marked in green (

**a**), and its boundary points and bisector points in blue (

**b**), relative to the helix with the specified location.

**Figure 23.**This schematic illustration of the shape of the Hermite arc of the third-order spatial curve.

**Figure 25.**Worm gear driving and the worm cutter [56].

**Figure 26.**The worm gear hob tooth with (

**a**) the cutting-edge curve marked with blue line and root cylinder curve marked with green line on the face surface H and the projector directions

**v**

_{1}and

**v**

_{2}; (

**b**) the angle of the chip groove ω

_{1}in addition to the usual notations [40].

**Figure 27.**Schematic sketch of the relationship between a Bezier curve and a Hermite arc interpolating to the same spatial curve.

**Figure 28.**Points of the surfaces bounding the bijective subsets of the Monge cuboid are marked in green for setting the hob, in blue for measuring the wear of the cutting edge, in red for a fixed groove angle.

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Balajti, Z.
Challenges of Engineering Applications of Descriptive Geometry. *Symmetry* **2024**, *16*, 50.
https://doi.org/10.3390/sym16010050

**AMA Style**

Balajti Z.
Challenges of Engineering Applications of Descriptive Geometry. *Symmetry*. 2024; 16(1):50.
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**Chicago/Turabian Style**

Balajti, Zsuzsa.
2024. "Challenges of Engineering Applications of Descriptive Geometry" *Symmetry* 16, no. 1: 50.
https://doi.org/10.3390/sym16010050