1. Introduction

2. Materials and Methods

3. Results

3.1. Surfaces’ intersection and flattening through locus - based parametric functions

3.1.1. Surface to surface intersections

The encounter or intersection curve between two 3D surfaces, ${\mathbf{S}}_{\mathbf{1}}$ and ${\mathbf{S}}_{\mathbf{2}}$, $\mathbf{C}$, may be defined as the set of all 3D points, P, that pertain to both surfaces ($\mathrm{P}\in {\mathbf{S}}_{\mathbf{1}}\cap {\mathbf{S}}_{\mathbf{1}}$).

We define $\mathbf{S}\left(\omega \right)$ as collection of auxiliary surfaces that intersects both with ${\mathbf{S}}_{\mathbf{1}}$ and ${\mathbf{S}}_{\mathbf{2}}$ when the argument $\omega \in \Omega$, where $\Omega$ is a reference set on ${\Re }^n$ for n = 1 or $n=2$. Accordingly, the point $\mathrm{P}\left(\omega \right)={\mathbf{C}}_{\mathbf{1}}\left(\omega \right)\cap {\mathbf{C}}_{\mathbf{2}}\left(\omega \right)$ will pertain to $\mathbf{C}$, for ${\mathbf{C}}_{\mathbf{1}}\left(\omega \right)={\mathbf{S}}_{\mathbf{1}}\cap \mathbf{S}\left(\omega \right)$ and ${\mathbf{C}}_{\mathbf{2}}\left(\omega \right)={\mathbf{S}}_{\mathbf{2}}\cap \mathbf{S}\left(\omega \right)$.

If we choose a reference set $\Omega$, such that the collection $\mathbf{S}(\omega \in \Omega )$ sweeps all the points from ${\mathbf{S}}_{\mathbf{1}}$ and ${\mathbf{S}}_{\mathbf{2}}$, then we may set the intersection curve, $\mathbf{C}$, as the set of points $\mathrm{P}\left(\omega \right)={\mathbf{C}}_{\mathbf{1}}\left(\omega \right)\cap {\mathbf{C}}_{\mathbf{2}}\left(\omega \right)$ for $\omega \in \Omega$.

The projection of $\mathrm{P}\left(\omega \right)$ from $\mathbf{C}$ onto a plane is called ${\mathrm{p}}_{\sigma }$($\omega$), in such a way that $\sigma$ is the projection curve of $\mathbf{C}$ on that plane. As we may completely define a 3D curve from any two non-degenerated plane projections [14], we may compute the intersection curve $\mathbf{C}$ through their orthogonal projections on the dihedral planes, $\sigma$ (horizontal) and ${\sigma }^{\prime }$ (vertical).

Descriptive geometry gives algebraic procedures that solve ${\mathrm{p}}_{\sigma }$ projection for any value of $\omega$. The encapsulation of that sequence of algebraic procedures through dependent algebraic objects for the parameter $\omega$ may achieve a reliable locus entity, defining the following equation for the horizontal projection curve, $\sigma$:

$${\mathcal{L}}_{\sigma }\left(\omega \right)\equiv \sigma =locus({\mathrm{p}}_{\sigma }\left(\omega \right),\omega \in \Omega ),$$

(2)

where ${\mathcal{L}}_{\sigma }$ is a 2D parametric function in $\Omega$, and locus is the algebraic entity that gives the points ${\mathrm{p}}_{\sigma }$ associated with the value $\omega \in \Omega$. The vertical projection, ${\sigma }^{\prime }$, is achieved by the same equation when the horizontal projection of $\mathrm{P}\left(\omega \right)$ is substituted by the vertical projection, ${\mathrm{p}\text{’}}_{\sigma }$. This method was applied successfully to build ${\mathrm{C}}_{\mathrm{E}}\mathrm{D}\mathrm{G}$ models using GeoGebra as dynamic geometry software [10].

Figure 3 presents the computation of the intersection between an oblique cone with an oblique cylinder using this method. They are defined by their circular sections with an horizontal plane (bases) and their axis. The vertical contour of the cone has been added to facilitate the interpretation.

We select the collection $\mathbf{S}\left(\omega \right)$ of planes that contain the cone vertex, v’ - v, and a line parallel to the cylinder axis, called r’ - r. As the incidence of this plane onto the horizontal plane is the h’ - h point, a plane from this collection may be defined through the s’ - s horizontal line, as shown in Figure 3. The line must intersect the two circular bases to assure that $\mathbf{S}\left(\omega \right)$ intersects the two surfaces. The lines h - PLI and h - PLS define the boundary planes of this collection for the relative position of the surfaces. These lines must be tangents to any of the circular bases. This manner, the PLS point is computed as PLS = If(y(1)<y(3) then point 1 else point 3), where 1 and 3 are presented in Figure 3 and y is the vertical coordinate. A similar condition with points 2 and 4 defines the PLI point. This conditional logic attends any relative position of these oblique surfaces.

We have defined $\omega$ as the angle ${\gamma }_{\mathrm{P}}$ between lines s and h - PLS (9.8° in the figure). Therefore the reference set, $\Omega$, is the interval [0, ${\gamma }_{\mathrm{Px}}$], where ${\gamma }_{\mathrm{Px}}$ is the angle between lines PLI - h - PLS (12.8° in the figure).

A general plane $\mathbf{S}\left(\omega \right)$ intersects in two generatrix lines (${\mathbf{C}}_{\mathbf{1}}\left(\omega \right)$) for the cylinder and other two for the cone (${\mathbf{C}}_{\mathbf{2}}\left(\omega \right)$), which intersects to gives four intersection points (${\mathrm{P}}_{\mathrm{i}}\left(\omega \right)={\mathbf{C}}_{\mathbf{1}}\left(\omega \right)\cap {\mathbf{C}}_{\mathbf{2}}\left(\omega \right),i=1,2,3,4$), from $\mathbf{C}$, as shown in Figure 3 (ac’ - ac, bc’ - bc, ad’ - ad, bd’ - bd) . The constructive procedure is shown. Therefore, the vertical projection of $\mathbf{C}$, ${\sigma }^{\prime }$, is given by the following four leaves parametric curve:

$${\sigma }^{\prime }\equiv {\mathcal{L}}_{{\sigma }^{\prime }}\left(\omega \right)=locus(\{a{c}^{\prime },b{c}^{\prime },a{d}^{\prime },b{d}^{\prime }\},\omega \in \Omega )$$

(3)

The leaves of ${\sigma }^{\prime }$ have been emphasized through alternating colours (see Figure 3). The horizontal projection of $\mathbf{C}$, $\sigma$, is also a four - leaves parametric curve given by

$$\sigma \equiv {\mathcal{L}}_{\sigma }\left(\omega \right)=locus(\{ac,bc,ad,bd\},\omega \in \Omega )$$

(4)

Figure 4. Projection curves of cone - cylinder intersection (penetration) in a ${\mathrm{C}}_{\mathrm{E}}\mathrm{D}\mathrm{G}$ model.

Figure 4. Projection curves of cone - cylinder intersection (penetration) in a ${\mathrm{C}}_{\mathrm{E}}\mathrm{D}\mathrm{G}$ model.

Reducing the distance between the centers of circular bases, o - ${\mathrm{o}}_2$, the bite between surfaces changes to a penetration of the cylinder into the cone, as shown in Figure 4. The boundary planes are now both tangent to the cylinder bases (PLS is 3 and PLI is 4). The intersection curve splits into two separates pieces, that are defined by the ${\mathcal{L}}_{{\sigma }^{\prime }}\left(\omega \right)$ parametric curve of equation (3) for the vertical projection and ${\mathcal{L}}_{\sigma }\left(\omega \right)$ of equation (4) for the horizontal projection.

An interesting property of this technique is that it keeps the geometric integrity of the intersection curve (i.e. these curves are not approximations but the true algebraic object). As Figure 3 and Figure 4 illustrate, it may be implemented in dynamic geometry tools through the graphical entities associated with the descriptive geometry’s algebraic objects. The input surfaces are not limited to ruled or quadric surfaces, but to smooth surfaces for which some computable auxiliary surface produces low complexity intersection curves. Most technical surfaces fulfil with these requirements.

3.1.2. Surface flattening

Ruled single curvature surfaces can be flattened without deformation. Here we present a locus based technique that gives the exact flat transformed of any 3D curve in this type of surfaces. It has been implemented in GeoGebra as part of the built-in features of ${\mathrm{C}}_{\mathrm{E}}\mathrm{D}\mathrm{G}$.

The flat transformation of a curve $\mathbf{C}$ that pertains to a ruled single curvature surface, $\mathbf{S}$, may be formally computed by means of a two-stage procedure. First, a generic generatrix line of $\mathbf{S}$, g, that contains a point $\mathrm{P}\in \mathbf{C}$ is transformed to the flat domain. This transformation is equivalent to the placement of a line in the flat domain. Secondly, the point P is transformed to the flat domain, that is, it is placed onto the g transformed. This two-stage procedure defines a sequence of dependent algebraic objects, which builds the locus entity that defines the flat transformed of $\mathbf{C}$ through a parametric function. Figure 5 shows a scheme of this process for a cylinder and a cone, both oblique with generic directrix.

In the case of a cylinder (Figure 5 - left), we must intersect it with a plane perpendicular to the axis, to give the section $\sigma$. The flat transformed of $\sigma$ is the R straight line, perpendicular to transformed generatrix lines. We may use any generatrix line as reference in the flat domain, as shown in Figure 5 (dashed line). Then, defining ${\mathrm{p}}_{\sigma }$$\in \sigma$ as a parameter, it is transformed to ${\mathrm{f}}_{\mathrm{R}}$(${\mathrm{p}}_{\sigma }$) onto R, assuring the invariance of lengths of transformed paths. That is, the distance between ${\mathrm{f}}_{\mathrm{R}}$(${\mathrm{P}}_{\sigma }$) and ${\mathrm{f}}_{\mathrm{R}}$(${\mathrm{p}}_{\sigma 0}$) is equal to the length of $\sigma$ between ${\mathrm{P}}_{\sigma }$ and ${\mathrm{P}}_{\sigma 0}$. The flat transformed of the generatrix line through ${\mathrm{p}}_{\sigma }$ is parallel to the reference one. The function ${\mathrm{f}}_{\mathrm{R}}$ converts its argument from $\sigma$ into the transformed in R.

Finally, the flat transformed of ${\mathrm{P}}_{\mathrm{A}}\in \mathcal{A}$ in the cylinder is ${\mathrm{P}}_{\pi }$, which pertains to the flat transformed of its generatrix line, with a distance between ${\mathrm{P}}_{\pi }$ and ${\mathrm{f}}_{\mathrm{R}}$(${\mathrm{P}}_{\sigma }$) equal to the distance between ${\mathrm{P}}_{\mathrm{A}}$ and ${\mathrm{P}}_{\sigma }$.

Considering that descriptive geometry provides mathematical procedures to achieve the invariance conditions and manipulate the 3D space through algebraic objects, the flat transformed of $\mathcal{A}$, $\pi$, may be computed as:

$$\pi \equiv {\mathcal{L}}_{\pi }\left(\omega \right)=locus(P_{\pi }\left(f_R\left(\omega \right)\right),\omega \in \Omega ),$$

(5)

where ${\mathcal{L}}_{\pi }$ is a 2D parametric function of $\omega ={\mathrm{P}}_{\sigma }\in \sigma$, $\Omega$ is a subset of $\sigma$ that defines the cylinder piece to be flattened, and $P_{\pi }\left(f_R\left(\omega \right)\right)$ is a point of $\pi$ defined by the generatrix line through $f_R\left(\omega \right)$.

In the case of an oblique cone as surface $\mathbf{S}$ to be flattened (Figure 5 - right), we first intersect this surface with a sphere centered in cone’s vertex (V) to obtain a warped curve in $\mathbf{S}$, $\mathcal{A}$, which is called support curve. We know that the flat transformed of $\mathcal{A}$ is a circular arch (D) with the radius of the sphere. The flat transformed of ${\mathrm{P}}_{\mathrm{A}}\in \mathcal{A}$ is ${\mathrm{f}}_{\mathrm{D}}\left({\mathrm{P}}_{\mathrm{A}}\right)\in \mathrm{D}$, with a distance to the reference point ${\mathrm{f}}_{\mathrm{D}}\left({\mathrm{P}}_{\mathrm{A}0}\right)$ along D equal to the distance of ${\mathrm{P}}_{\mathrm{A}}$ to ${\mathrm{P}}_{\mathrm{A}0}$ along $\mathcal{A}$ (see Figure 5). The flat transformed of a generatrix line through ${\mathrm{P}}_{\mathrm{A}}\in \mathcal{A}$ contains the flat transformed of vertex (${\mathrm{V}}_{\mathrm{ref}}$ is placed together with the reference generatrix line) and ${\mathrm{f}}_{\mathrm{D}}\left({\mathrm{P}}_{\mathrm{A}}\right)$.

The flat transformed of ${\mathrm{P}}_{\mathrm{C}}\in \mathbf{C}$ in the cone is ${\mathrm{P}}_{\tau }$, which pertains to the flat transformed of its generatrix line, with a distance between ${\mathrm{P}}_{\tau }$ and ${\mathrm{f}}_{\mathrm{D}}$(${\mathrm{P}}_{\mathrm{A}}$) equal to the distance between ${\mathrm{P}}_{\mathrm{C}}$ and ${\mathrm{P}}_{\mathrm{A}}$. Under the same consideration of equation 5, the flat transformed of $\mathbf{C}$ ($\tau$) may be computed as:

$$\tau \equiv {\mathcal{L}}_{\tau }\left(\omega \right)=locus(P_{\tau }\left(f_D\left(\omega \right)\right),\omega \in \Omega ),$$

(6)

where ${\mathcal{L}}_{\tau }$ is a 2D parametric function of $\omega \in \Omega$, such that the generatrix line through $\omega$ contains ${\mathrm{P}}_{\mathrm{A}}$, and $\Omega$ is a plane section of the cone that facilitates the algebraic manipulation of this surface.

The variable $\omega \in \Omega$ may be substituted by other variable ${\mathrm{t}}_{\omega }$$\in \mathcal{T}$ , if there exists any mapping function between $\Omega$ and $\mathcal{T}$.

The computation of the distance from ${\mathrm{P}}_{\mathrm{A}}$ to ${\mathrm{P}}_{\mathrm{A}0}$ along $\mathcal{A}$ is not trivial since $\mathcal{A}$ is a warped curve with a geometry that depends on the conical surface. We use the flat transformed of $\mathcal{A}$ as pertaining to a cylinder, taking advantage of the invariance of lengths during the flat transformation.

This method for surface flattening is applied to the conical surface intersected with the cylinder (bite) from the previous ${\mathrm{C}}_{\mathrm{E}}\mathrm{D}\mathrm{G}$ model, between two selected generatrix lines, as presented in Figure 6.

According to our nomenclature, projections are presented in lowercase, whereas flat transformations are in uppercase (excepting for Greek symbols). In Figure 6, ${\omega }_{\mathrm{R}}$ is a point in $\Omega$, which is the circular arch from ${\mathrm{r}}_{\mathrm{B}}$ to ${\mathrm{r}}_{\mathrm{A}}$ in the circular base of the conical surface (plane section). It is defined as ${\omega }_{\mathrm{R}}$ = Point(circular base, ${\mathrm{t}}_{\omega }\mathrm{R}$), where ${\mathrm{t}}_{\omega }\mathrm{R}$ is a real parameter in $\mathcal{T}=[0,1]$ and Point is the mapping function between $\Omega$ and $\mathcal{T}$.

The generatrix line through ${\omega }_{\mathrm{R}}$ intersects to the sphere (radius equal to segment v’ - ${\mathrm{r}\text{’}}_{\mathrm{A}}$) in ${\mathrm{p}\text{’}}_{\mathrm{AR}}$ - ${\mathrm{p}}_{\mathrm{AR}}$. Accordingly, ${\mathrm{p}}_{\mathrm{AR}}$ pertains to the horizontal projection of the support curve $\mathcal{A}$, which is calculated through equation (2) using ${\omega }_{\mathrm{c}}$ in the circular base, defined between ${\mathrm{r}}_{\mathrm{A}}$ and ${\omega }_{\mathrm{R}}$. Therefore, the horizontal projection of $\mathcal{A}$ between ${\mathrm{r}}_{\mathrm{A}}$ and ${\omega }_{\mathrm{R}}$ is locus(${\mathrm{p}}_{\mathrm{Ac}}$, ${\omega }_{\mathrm{c}}$) (orange horizontal projection).

We define a right cylinder from the horizontal projection of $\mathcal{A}$ (bottom base) to the support curve (warped) $\mathcal{A}$. The flat transformed of $\mathcal{A}$ as a curve of the cylinder is $\pi$, which is shown in Figure 6 (upper - right). According to the equation (5), $\pi$ is defined as locus(${\mathrm{P}}_{\mathrm{AR}}$, ${\mathrm{t}}_{\omega }\mathrm{R}$). The distance between ${\mathrm{r}}_{\mathrm{A}}$ and ${\mathrm{p}}_{\mathrm{ARL}}$ in the horizontal projection of $\mathcal{A}$ is computed through Perimeter(locus) command.

The distance between ${\mathrm{R}}_{\mathrm{A}}$ and ${\mathrm{P}}_{\mathrm{AR}}$ along D (circular arch in cone flat domain, Figure 6 bottom - right) must be equal to the distance between ${\mathrm{R}}_{\mathrm{A}}$ and ${\mathrm{P}}_{\mathrm{AR}}$ along $\pi$ curve, which is known. Therefore, ${\mathrm{P}}_{\mathrm{AR}}$ in D and its associated generatrix line through $\left({\omega }_{\mathrm{R}}\right)$ may be placed in the flat domain. The flat transformation of the cone base through ${\mathrm{R}}_{\mathrm{A}}$ to ${\mathrm{R}}_{\mathrm{B}}$ may be computed as locus((${\omega }_{\mathrm{R}}$), ${\mathrm{t}}_{\omega }\mathrm{R}$), according equation (6). Finally, the flat transformation of the cone - cylinder intersection inside the flat pattern is defined by two parametric functions: locus(${\omega }_{\mathrm{R}}$F, ${\mathrm{t}}_{\omega }\mathrm{R}$) and locus(${\omega }_{\mathrm{R}}$E, ${\mathrm{t}}_{\omega }\mathrm{R}$), since ${\omega }_{\mathrm{R}}$f’ - ${\omega }_{\mathrm{R}}$f and ${\omega }_{\mathrm{R}}$e’ - ${\omega }_{\mathrm{R}}$e are the intersections of the generatrix line through ${\omega }_{\mathrm{R}}$ with the cylinder.

The ${\mathrm{C}}_{\mathrm{E}}\mathrm{D}\mathrm{G}$ model updates dynamically to changes of the parameters, what includes the modification of the intersection. This manner, if we reduce the distance between the centers of circular bases o - ${\mathrm{o}}_2$, as was performed in Figure 4, the bite will change to penetration, splitting in two pieces, and the flat pattern dynamically changes to give what shows Figure 7. It must be noted that we have also moved ${\mathrm{r}}_{\mathrm{B}}$ to increase the cone surface that is flattened.

Figure 7. Cone surface intersected with cylinder (penetrated, Figure 4) flattened between generatrix lines through ${\mathrm{r}}_{\mathrm{A}}$ and ${\mathrm{r}}_{\mathrm{B}}$.

Figure 7. Cone surface intersected with cylinder (penetrated, Figure 4) flattened between generatrix lines through ${\mathrm{r}}_{\mathrm{A}}$ and ${\mathrm{r}}_{\mathrm{B}}$.

4. Comparative analysis and discussion

• Feasibility to reach the required models. The 3D model of the hopper, which include the ducts connections were properly obtained both in ${\mathrm{C}}_{\mathrm{E}}\mathrm{D}\mathrm{G}$ and CAD, as shown in Figure 13 and Figure 19. Nonetheless, Solid Edge 2023 was not able to compute the flat pattern of the lower duct (truncated cone) because this duct encounters to the oblique cylindrical duct with an intersection of bite type. We used different strategies, as described in the Results section, without success.
• Once the 3D models were computed, we tried to use them for the analysis of the influence of the geometrical parameters in the outlet / inlet area ratio of the fluid duct, ${\mathrm{A}}_{\mathrm{r}}$, and finally for the optimization of the hopper to achieve ${\mathrm{A}}_{\mathrm{r}}\ge 3$ in a fast expansion. The ${\mathrm{C}}_{\mathrm{E}}\mathrm{D}\mathrm{G}$ model allowed a visual inspection of the fluid duct - upper duct connection through spatial rotation, as well as the plotting and quantitative computation of the relationship between ${\mathrm{A}}_{\mathrm{r}}$ and any geometrical parameter of the 3D system. We used this feature to plot ${\mathrm{A}}_{\mathrm{r}}$(Ecc, Con) and select the design values Ecc = 1.66 m and Con = 0.09 (Figure 12b). In opposition, the model computed with Solid Edge 2023 does not provide a direct manner to extract the ${\mathrm{A}}_{\mathrm{r}}$ function.
• With respect to the accuracy, the comparison between Table 2 and Table 3 shows that the position of ${\mathrm{P}}_3$ - ${\mathrm{P}}_4$ (boundary points) in the flat pattern had relative deviations lesser than 0.01 %. In the case of ${\mathrm{P}}_{\mathrm{m}}$ and ${\mathrm{P}}_{\mathrm{M}}$, z relative deviations were lesser than 0.02 %, whereas y relative deviations were lesser than 3.9 %. The relative deviations between ${\mathrm{A}}_{\mathrm{r}}$ values were lesser than 0.6 % with the exception of the value for ${\mathrm{Nom}}_0$ dimensions’ group which was 8.7 %. Values greater than 5 % occurred in those cases where a manual selection of some 3D object were needed. We conclude that the accuracy was high in both models.

Abbreviations

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