# A 3D Descriptive Geometry Problem-Solving Methodology Using CAD and Orthographic Projection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- Orthogonal views.
- Three-dimensional definition and creation of the plane main lines.
- Defining and creating views to show the true magnitude of lines and plane figures.
- Identifying, defining, and creating lines: intersecting, parallel, and perpendicular.
- Determining the angle between two coplanar lines, line and plane, and two planes.

#### 2.1. Orthographic Projection Fundamentals

_{n}, Y

_{n}, Z

_{n}), n ∈ [1, 6]. Thus, the top view corresponds to the X

_{n}, Y

_{n}, the front view to X

_{n}, Z

_{n}, and the side view to the Y

_{n}, Z

_{n}coordinates.

#### 2.2. CAD Fundamentals

#### 2.2.1. Reference System

- World and User Coordinate Systems

- Reference Plane

- Coordinate system

#### 2.2.2. Viewpoint

#### 2.2.3. User Coordinate System

#### 2.2.4. Viewports

#### 2.3. CAD Tools

#### 2.3.1. Rotation

#### 2.3.2. First-Angle and Third-Angle Projection

#### 2.3.3. Point Filters

## 3. Results

#### 3.1. Orthogonal Views

#### 3.2. Identifying, Defining, and Creating Principal Lines of a Plane

#### 3.3. True Length of a Line and True Size of a Plane

#### 3.4. Parallelism and Perpendicularity

#### 3.4.1. Parallelism between Lines

#### 3.4.2. Parallelism between Line and Plane

#### 3.4.3. Parallelism between Planes

#### 3.4.4. Perpendicularity between Lines

#### 3.4.5. Perpendicular Line from a Point to a Line

#### 3.4.6. Perpendicular Line and Shortest Distance between Two Skew Lines

- CADOP vs. traditional method.

- CADOP solving procedure step by step.

- The plane is defined through line b parallel to line a (Figure 26). Line a1 is a copy of line a through any point P on line b. The plane is determined by obtaining the UCS formed by line a1 and b.

- 2.
- Making Zu = 0 for any two points on line a results in a projection onto the UCS defined before. Figure 27 shows line ap resulting from the projection.

- 3.
- The perpendicular line and shortest distance segment MN between lines a and b are obtained. As shown in Figure 28, the intersection of the projected line ap and line b determines point N which is one of the ends of the solution segment. The other end, point M, is the intersection of line a with the line that, drawn through N, is perpendicular to the plane defined by the UCS. Segment MN is parallel to the Z axis of this UCS, and its true length is obtained directly from the information provided by CAD.

- 4.
- Finally, the top and front views and true length dimension of segment MN are obtained. Figure 30 shows the solution provided by applying CADOP.

#### 3.5. Angles

#### 3.5.1. Angle between Two Coplanar Lines

#### 3.5.2. Angle between Line and Plane

#### 3.5.3. Angle between Two Planes

#### 3.6. Regular Polyhedra

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**User coordinate system definition: (

**a**) XY plane through three points and (

**b**) Z axis direction through two points.

**Figure 8.**Isometric view at a 5:1 scale, and front and top views at a 2:1 scale of a solid shown in three different viewports.

**Figure 10.**Different conventions for arranging views: first-angle projection (

**top**); third-angle projection (

**bottom**).

**Figure 11.**(

**a**) Orthographic projections of a point; front view coordinates X, Z and top view coordinates X, Y. (

**b**) Front view rotated 90° with respect to the X axis. (

**c**) Application to an oblique line defined by points A and B.

**Figure 14.**(

**a**) Top and front views of line AB. (

**b**) Restoring the spatial position of line AB from its horizontal and vertical projections.

**Figure 16.**Principal lines through a point of a plane using the intersection of surfaces: (

**a**) horizontal line h through point A and (

**b**) frontal line f through point B.

**Figure 17.**True size of the plane ABCD by (

**a**) rotating the body without modifying the coordinate system; and (

**b**) considering an UCS.

**Figure 19.**Using the UCS for checking parallelism between lines. Lines m and t are not parallel, while lines r and s are.

**Figure 22.**Determining the perpendicular line and shortest distance between two skew lines, r and s, applying the traditional method. A sketch of the followed procedure is shown on the bottom right corner.

**Figure 23.**Solution obtained using CADOP for the perpendicular line and shortest distance between two skew lines r and s. (

**a**) Hiding the layers in which intermediate steps are drawn. (

**b**) Arranging the views for avoiding projections overlapping. (

**c**) Three-dimensional view.

**Figure 29.**Three-dimensional position of perpendicular line and shortest distance segment MN between skew lines a and b.

**Figure 31.**Angle between two coplanar lines r and s: (

**a**) top and front view of r and s; (

**b**) matching UCS to the plane defined by r and s; and (

**c**) angle φ measurement in the top view of the UCS.

**Figure 32.**Angle between line and plane: angle φ measurement in the top view of the UCS defined by line r and its projection rp onto a plane.

**Figure 33.**Angle φ between two planes π and ω: (

**a**) UCS Z axis matches to the intersection line of both planes; (

**b**) angle φ measurement is shown in top view.

**Figure 34.**Regular polyhedra problem solved with CADOP: (

**a**) tetrahedron obtained in 3D, knowing that two of its opposite edges are located on two perpendicular skew lines; (

**b**) top and front views of the polyhedron solution.

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**MDPI and ACS Style**

Gutiérrez de Ravé, E.; Jiménez-Hornero, F.J.
A 3D Descriptive Geometry Problem-Solving Methodology Using CAD and Orthographic Projection. *Symmetry* **2024**, *16*, 476.
https://doi.org/10.3390/sym16040476

**AMA Style**

Gutiérrez de Ravé E, Jiménez-Hornero FJ.
A 3D Descriptive Geometry Problem-Solving Methodology Using CAD and Orthographic Projection. *Symmetry*. 2024; 16(4):476.
https://doi.org/10.3390/sym16040476

**Chicago/Turabian Style**

Gutiérrez de Ravé, Eduardo, and Francisco J. Jiménez-Hornero.
2024. "A 3D Descriptive Geometry Problem-Solving Methodology Using CAD and Orthographic Projection" *Symmetry* 16, no. 4: 476.
https://doi.org/10.3390/sym16040476