< GeoGebra

This learning resource is about perpective drawing of 3D objects on a mirror and learn about geometric principles of construction. Together with this learning resource a GitHub repository with Geogebra Files[1] was created that can be opened with dynamic geometry software Geogebra. The construction is based on a german article in the journal Mathematica Didactica 26 (2003) Bd.1 43, Experimenteller Umgang mit Spiegelung und Perspektive unter Verwendung von Dynamischer Geometriesoftware, Engelbert Niehaus[2]. You can download the Geogebra files as ZIP from the GitHub repository[3].

3D Construction of Projections in Geogebra

The repository contains Geogebra files for learning 3D projections on a mirror Z-, X, and I-construction of 3D point and lines. This repository has additional learning material for the Open Source software Geogebra and the file are bundled in a GitHub repositopry[1] created for this Wikiversity Learning Resource about Perspective Drawing. Keep in mind that line through the point Z in the upper paper plane of the 3D world provides the correponding vanishing point directly by the intersection with the horizon h. Parallel lines in the 3D world have the same vanishing point on the horzion. This geometric property leads to the Z-construction to find the corresponding vanishing point for half lines in the ground plane.

Basic Situation in Front of a Mirror

The following image shows the basic situation of a 3D projection on a mirror. The perspective image of the box is painted on the mirror.

Use a Paper as Mediator between 3D and 2D Construction

You can use a paper as mediator between the 3D world and the 2D projection on the paper. The point Z is the point where the eye is located. h is the horizon of the contruction and s is parallel to the horizon h as the intersection of the groundplane and plane of the mirror. Projected objects "stand" on the groundplane and the projection to the mirror plane is the objective of the construction.

Basic Geometry in Front of the Mirror

The perspective drawing creates an image of an object in front of a mirror. The eye observes the projective image from a point Z. Assume the observer is painting the perspective image of a vertical line/stick on the surface of the mirror. Considering the perpective construction from the side will lead to the following situation.

Z-Construction for Lines on a Surface

We unfold the paper mediator between 2D and 3D construction and transfer the folding to two parallel lines in Geogebra. A Z-construction generates perspective image of a half-line b with ground line s. The intersection with the ground line is the point S1. Assume you want to draw the projective image of the half line b on the mirror plane. To create the projective image you need the following steps.

  • draw a parallel line d to the half line b through the point of the eye Z. The intersection of that line d with the horizon generates the vanishing point F2 for the projection of the halfline b to the mirror plane.
  • The line between F1 and S1 is the projective image of the half-line b.


X-Construction for Point on a Surface

The X-construction creates the perspective image of a point by using 2 Z-constructions.

I-Construction for a Vertical Line

The I-construction creates the perspective image of vertical line segment by using 2 X-constructions.

3D-Object generated with Z-,X- and I-Construction

External Resources

See also

References

  1. 1 2 3D Construction with Geogebra (2017-2019) Engelbert Niehaus, URL: https://github.com/niebert/3D_Construction_Geogebra (accessed 2019/06/12)
  2. ↑ Niehaus, Engelbert (2003), Experimenteller Umgang mit Spiegelung und Perspektive unter Verwendung von Dynamischer Geometriesoftware, Mathematica Didactica 26 (2003) Bd.1 43, http://www.mathematica-didactica.com/altejahrgaenge/md_2003/md_2003_1_Niehaus_Spiegelung.pdf
  3. ↑ Gegeobra Files as ZIP - 3D Construction (2017-19) URL: https://github.com/niebert/3D_Construction_Geogebra/archive/master.zip - (accessed 2020/11/27)
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