# GeoGebra/Perspective Drawing on Mirror

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## Introduction[edit | edit source]

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]}.

## Learning Tasks[edit | edit source]

The tasks are divided into the following areas.

- Theoretical background
- Geogebra construction
- Projection of circles
- Projection of a sine curve

### Theoretical background[edit | edit source]

Analyze the theoretical background^{[2]} for perspective drawing^{[4]}, in which the non-digital environment is a mirror used as a projection surface on which, for example, the perspective image is drawn with a foil pen.

#### Vanishing Point Perspective[edit | edit source]

The Vanishing point perspective is a way of perspective drawing.

#### Perspective Drawing in Art[edit | edit source]

### Geogebra construction[edit | edit source]

Starting from this situation of drawing objects in perspective on a mirror, we will now discuss constructions in Geogebra:

*Z*construction for the projection of half-lines,*X*construction for the projection of points as the intersection of two half-lines. Here, the image point is generated via two*Z*constructions.*I*construction for the projection of a perpendicular line generated via two*X*constructions.

### Projection of circles[edit | edit source]

Consider how to use a locus line and an *X* construction to create the projective image of a circle.

### Projection from a sine curve[edit | edit source]

Consider how to use a locus line and an *X* construction to create the projective image of a sine curve in the standing plane. First, draw the graph of the figure:

for appropriate parameters . Use sliders for these parameters. Use the X-construction to the create perspective projection of a single point on the graph of

## 3D Construction of Projections in Geogebra[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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

## Z-Construction for Lines on a Surface[edit | edit source]

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[edit | edit source]

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

## I-Construction for a Vertical Line[edit | edit source]

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[edit | edit source]

## Stereoscopy[edit | edit source]

Spatial perception is required to interact with your environment and relevant for geometric and spatial learning objectives (e.g. in chemistry^{[5]}. The constructions mentioned above can be extended to Stereoscopy.

### Learning Activity - Projection of Circle[edit | edit source]

Use trigonometric functions to move a point on a circle with a slider. Use the X-construction to create the corresponding projected point and move the a slider between the value in the plane of the stand on circular path.

### Animation for the Projection of Circles[edit | edit source]

The following GIF animation shows the projection of a circle with a slider for the angle and application of the X-construction to generate the perspective trace of the projected point of the given circle in the plane of stand.

### Learning Activities - Stereoscopy[edit | edit source]

- Analyze the stereoscopic image for spatial differences in the projective mapping for the left and right eye.
- Describe the applications of Stereoscopy and history when the first applications Stereoscopy were used.

### Animation - Distance of Eyes[edit | edit source]

The following animation shows the difference of the projections according to the position of left eye and right eye .

## External Resources[edit | edit source]

- GitHub Resources with Geogebra Files as ZIP-File
- GitHub Repository for Wikiversity Learning Resource
- Orthogonal Projection in Schools with Geogebra - by Anna Hundertmark

## See also[edit | edit source]

- 3D Modelling
- Projective Geometry
- Regard3D
- Photogrammetry
- Perspective Drawing
- Equirectangular projection
- Stereoscopy

## References[edit | edit source]

- ↑
^{1.0}^{1.1}3D Construction with Geogebra (2017-2019) Engelbert Niehaus, URL: https://github.com/niebert/3D_Construction_Geogebra (accessed 2019/06/12) - ↑
^{2.0}^{2.1}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 - ↑ Gegeobra Files as ZIP - 3D Construction (2017-19) URL: https://github.com/niebert/3D_Construction_Geogebra/archive/master.zip - (accessed 2020/11/27)
- ↑ D'amelio, J. (2004). Perspective drawing handbook. Courier Corporation.
- ↑ Coleman, S. L., & Gotch, A. J. (1998). Spatial perception skills of chemistry students. Journal of Chemical Education, 75(2), 206.

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