# Intersection and Flattening of Surfaces in 3D Models through Computer-Extended Descriptive Geometry (CeDG)

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- Feasibility to reach the parametric 3D hopper’s model and the required flat patterns.
- Application of the 3D model to achieve ${\mathrm{A}}_{\mathrm{r}}$ > 3 with minimum conicity through the eccentricity dimension.
- Accuracy of the ${\mathrm{D}}_{\mathrm{u}}$ and ${\mathrm{D}}_{\mathrm{l}}$ flat patterns.

## 3. Results

#### 3.1. Surfaces’ Intersection and Flattening through Locus-Based Parametric Functions

#### 3.1.1. Surface-to-Surface Intersections

#### 3.1.2. Surface Flattening

`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 $\mathcal{T}$.

`Perimeter`(locus) command.

#### 3.2. Hopper’s CeDG Modeling

#### 3.3. Hopper’s CAD Modeling

## 4. Comparative Analysis and Discussion

- Feasibility to reach the required models. The 3D model of the hopper that includes the ducts connections was properly obtained both in CeDG 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 the oblique cylindrical duct with an intersection of the bite type. We used different strategies, as described in the Section 3, 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 CeDG 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, we did not find a direct manner to extract the ${\mathrm{A}}_{\mathrm{r}}$ function in Solid Edge 2023.
- With respect to accuracy, a 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 less than 0.01%. In the case of ${\mathrm{P}}_{\mathrm{m}}$ and ${\mathrm{P}}_{\mathrm{M}}$, z relative deviations were lower than 0.02%, whereas y relative deviations were lower than 3.9%. The relative deviations between ${\mathrm{A}}_{\mathrm{r}}$ values were lower 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 was needed. We conclude that the accuracy was high in both models.

`Locus`implicit function cannot be used as a general 2D geometric object, and the measurement of lengths along it has limitations [16]. These issues define other ongoing research lines.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${\mathbf{S}}_{\mathbf{i}}$ | Data surfaces for $i=1,2$ |

$\mathbf{C}$ | Curve in 3D space |

$\overline{AB}$ | Distance between A and B points |

A (a’ − a) | Spatial (3D) object (vertical projection—horizontal projection) |

${\mathrm{P}}_{\mathrm{y}}$ | horizontal distance between P and right A points in flat pattern (Figure 10) |

${\mathrm{P}}_{\mathrm{z}}$ | vertical distance between P and right A points in flat pattern (Figure 10) |

CeDG | Computer extended Descriptive Geometry |

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**Figure 6.**Cone surface intersecting with a cylinder (bite, Figure 3) flattened between generatrix lines through ${\mathrm{r}}_{\mathrm{A}}$ and ${\mathrm{r}}_{\mathrm{B}}$. The projections of surfaces include auxiliary lines that help to follow the building process that gives the flat pattern (right–bottom).

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

**Figure 8.**CeDG modeling of ${\mathrm{D}}_{\mathrm{u}}$ − ${\mathrm{D}}_{\mathrm{f}}$ connection curve.

**Figure 9.**CeDG modeling of ${\mathrm{D}}_{\mathrm{l}}$ − ${\mathrm{D}}_{\mathrm{o}}$ connection curve.

**Figure 12.**Visual inspection of the fluid duct connection (

**a**) and numerical relation of its area section, ${\mathrm{A}}_{\mathrm{r}}$, as a function of eccentricity (abscissa) and conicity (0.09–0.25 in blue, 0.27–0.35 in red, increments of 0.02) (

**b**) in the CeDG model.

**Figure 14.**Solid Edge modeling of the ${\mathrm{D}}_{\mathrm{u}}$ − ${\mathrm{D}}_{\mathrm{f}}$ ducts system.

**Figure 15.**Solid Edge modeling of the ${\mathrm{D}}_{\mathrm{u}}$ duct with the ${\mathrm{D}}_{\mathrm{f}}$ connection curve.

**Figure 16.**Flat pattern of the ${\mathrm{D}}_{\mathrm{u}}$ cylinder together with the intersection curve produced by the ${\mathrm{D}}_{\mathrm{f}}$ truncated cone in the Solid Edge model.

**Figure 17.**Solid Edge modeling of the ${\mathrm{D}}_{\mathrm{l}}$ − ${\mathrm{D}}_{\mathrm{o}}$ ducts system.

Dim. Group | Con ^{†} | Ecc | ${\mathbf{D}}_{\mathbf{df}}$ | $\mathit{\alpha}$_{df} ^{‡} | $\mathit{\alpha}$_{do} ^{‡} |
---|---|---|---|---|---|

${\mathrm{Nom}}_{0}$ | 0.27 | 0.6 | 2 | 65${}^{\circ}$ | 45${}^{\circ}$ |

${\mathrm{Nom}}_{1}$ | 0.09 | 1.66 | 2 | 65${}^{\circ}$ | 45${}^{\circ}$ |

${\mathrm{Var}}_{0}$ | 0.5 | 1.11 | 0.5 | 50${}^{\circ}$ | 52${}^{\circ}$ |

${\mathrm{Var}}_{1}$ | 0.5 | 0 | 0.5 | 50${}^{\circ}$ | 52${}^{\circ}$ |

${\mathrm{Var}}_{2}$ | 0.09 | 2.43 | 0.5 | 50${}^{\circ}$ | 52${}^{\circ}$ |

^{†}Dimensionless.

^{‡}Sexagesimal degrees.

**Table 2.**Coordinates of selected points of flat patterns from upper and lower duct (m) and ${\mathrm{A}}_{\mathrm{r}}$ ratio, computed with CeDG for each dimension’s groups.

Dim. Group | ${\mathbf{P}}_{3\mathbf{y}}$ | ${\mathbf{P}}_{3\mathbf{z}}$ | ${\mathbf{P}}_{4\mathbf{z}}$ | ${\mathbf{P}}_{\mathbf{my}}$ | ${\mathbf{P}}_{\mathbf{mz}}$ | ${\mathbf{P}}_{\mathbf{My}}$ | ${\mathbf{P}}_{\mathbf{Mz}}$ | ${\mathbf{Q}}_{5\mathbf{g}}$ | ${\mathbf{Q}}_{5\mathit{\alpha}}$^{‡} | ${\mathbf{A}}_{\mathbf{r}}$^{†} |
---|---|---|---|---|---|---|---|---|---|---|

${\mathrm{Nom}}_{0}$ | 3.487 | 3.275 | 3.691 | 1.979 | 1.563 | 1.633 | 4.921 | 5.440 | 51.728${}^{\circ}$ | 3.259 |

${\mathrm{Nom}}_{1}$ | 3.657 | 3.147 | 4.269 | 2.656 | 2.016 | 1.637 | 4.777 | 5.440 | 39.679${}^{\circ}$ | 3.024 |

${\mathrm{Var}}_{0}$ | 3.478 | 2.276 | 4.010 | 2.554 | 0.972 | 0.977 | 4.268 | 7.464 | 44.246${}^{\circ}$ | 56.361 |

${\mathrm{Var}}_{1}$ | 2.287 | 2.476 | 2.476 | 1.124 | 0.881 | 1.098 | 3.668 | 7.464 | 62.302${}^{\circ}$ | 31.696 |

${\mathrm{Var}}_{2}$ | 2.110 | 2.642 | 4.177 | 1.901 | 2.320 | 0.963 | 4.227 | 7.464 | 38.599${}^{\circ}$ | 15.825 |

^{†}Dimensionless.

^{‡}Sexagesimal degrees.

**Table 3.**Coordinates of selected points of flat patterns from upper duct (m) and ${\mathrm{A}}_{\mathrm{r}}$ ratio, computed with Solid Edge 2023 for each dimension’s groups.

Dim. Group | ${\mathbf{P}}_{3\mathbf{y}}$ | ${\mathbf{P}}_{3\mathbf{z}}$ | ${\mathbf{P}}_{4\mathbf{z}}$ | ${\mathbf{P}}_{\mathbf{my}}$ | ${\mathbf{P}}_{\mathbf{mz}}$ | ${\mathbf{P}}_{\mathbf{My}}$ | ${\mathbf{P}}_{\mathbf{Mz}}$ | ${\mathbf{Q}}_{5\mathbf{g}}$ | ${\mathbf{Q}}_{5\mathit{\alpha}}$^{‡} | ${\mathbf{A}}_{\mathbf{r}}$^{†} |
---|---|---|---|---|---|---|---|---|---|---|

${\mathrm{Nom}}_{0}$ | 3.487 | 3.275 | 3.691 | 2.022 | 1.563 | 1.675 | 4.921 | - | - | 3.543 |

${\mathrm{Nom}}_{1}$ | 3.658 | 3.147 | 4.270 | 2.666 | 2.015 | 1.632 | 4.777 | - | - | 3.003 |

${\mathrm{Var}}_{0}$ | 3.478 | 2.276 | 4.009 | 2.551 | 0.971 | 0.981 | 4.268 | - | - | 56.618 |

${\mathrm{Var}}_{1}$ | 2.287 | 2.476 | 2.476 | 1.143 | 0.881 | 1.143 | 3.669 | - | - | 31.694 |

${\mathrm{Var}}_{2}$ | 2.111 | 2.642 | 4.177 | 1.831 | 2.312 | 0.761 | 4.228 | - | - | 15.930 |

^{†}Dimensionless.

^{‡}Sexagesimal degrees.

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**MDPI and ACS Style**

Prado-Velasco, M.; García-Ruesgas, L.
Intersection and Flattening of Surfaces in 3D Models through Computer-Extended Descriptive Geometry (CeDG). *Symmetry* **2023**, *15*, 984.
https://doi.org/10.3390/sym15050984

**AMA Style**

Prado-Velasco M, García-Ruesgas L.
Intersection and Flattening of Surfaces in 3D Models through Computer-Extended Descriptive Geometry (CeDG). *Symmetry*. 2023; 15(5):984.
https://doi.org/10.3390/sym15050984

**Chicago/Turabian Style**

Prado-Velasco, Manuel, and Laura García-Ruesgas.
2023. "Intersection and Flattening of Surfaces in 3D Models through Computer-Extended Descriptive Geometry (CeDG)" *Symmetry* 15, no. 5: 984.
https://doi.org/10.3390/sym15050984