View source for GeoGebra/Perspective Drawing on Mirror

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== Introduction ==
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 [https://github.com/niebert/3D_Construction_Geogebra GitHub repository with Geogebra Files]<ref name="gitrepo2003">3D Construction with Geogebra (2017-2019) Engelbert Niehaus, URL: https://github.com/niebert/3D_Construction_Geogebra (accessed 2019/06/12)</ref> was created that can be opened with dynamic geometry software [[w:en:Geogebra|Geogebra]]. The construction is based on a german article in the journal [http://www.mathematica-didactica.com/altejahrgaenge/md_2003/md_2003_1_Niehaus_Spiegelung.pdf Mathematica Didactica 26 (2003) Bd.1 43, Experimenteller Umgang mit Spiegelung und Perspektive unter Verwendung von Dynamischer Geometriesoftware, Engelbert Niehaus]<ref name="mathdid2003">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</ref>. You can download the [https://github.com/niebert/3D_Construction_Geogebra/archive/master.zip Geogebra files as ZIP from the GitHub repository]<ref>Gegeobra Files as ZIP - 3D Construction (2017-19) URL: https://github.com/niebert/3D_Construction_Geogebra/archive/master.zip - (accessed 2020/11/27)</ref>.

== Learning Tasks ==
The tasks are divided into the following areas.
* Theoretical background
* [[w:en:Geogebra|Geogebra]] construction
* Projection of circles
* Projection of a sine curve

=== Theoretical background=== 
Analyze the [http://mathematica-didactica.com/altejahrgaenge/md_2003/md_2003_1_Niehaus_Spiegelung.pdf theoretical background]<ref name="mathdid2003"/> for perspective drawing<ref>D'amelio, J. (2004). Perspective drawing handbook. Courier Corporation.</ref>, 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 ====
The [[w:en:Vanishing point|Vanishing point perspective]] is a way of [[w:en:Perspective_(graphical)|perspective drawing]].

[[File:08913-Perspective Run.jpg|450px|Shot from different distances]]

==== Perspective Drawing in Art ====

[[File:A Summer Day with perspective lines.jpg|450px|Perspective Lines in Paintings]]

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

=== Projection from a sine curve ===
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:
:<math> f(x) = a\cdot \sin(x-b) + c</math>
for appropriate parameters <math>a,b,c \in \mathbb{R}</math>. Use sliders for these parameters. Use the X-construction to the create perspective projection of a single point on the graph of <math>f</math>

== 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 [[w:en:Geogebra|Geogebra]] and the file are bundled in a [https://www.github.com/niebert/3D_Construction_Geogebra  GitHub repositopry]<ref name="gitrepo2003" /> created for this Wikiversity Learning Resource about Perspective Drawing. Keep in mind that line through the point <code>Z</code> in the upper paper plane of the 3D world provides the correponding vanishing point directly by the intersection with the horizon <code>h</code>. 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.

[[File:3d_object_perspective_drawing_red_lines.png|450px|Mirror-Basic]]

== 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 <code>Z</code> is the point where the eye is located. <code>h</code> is the horizon of the contruction and <code>s</code> is parallel to the horizon <code>h</code> 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.

[[File:paper_location_mirror_unfold.png|450px|Paper location as mediator between 3D and 2D world of projection ]]

== 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. 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. 

[[File:Sideview_i_construction_en.png|450px|Projection on Mirror]]

== 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 <code>b</code> with ground line <code>s</code>. The intersection with the ground line is the point <code>S1</code>. Assume you want to draw the projective image of the half line <code>b</code> on the mirror plane. To create the projective image you need the following steps. 
* draw a parallel line <code>d</code> to the half line <code>b</code> through the point of the eye <code>Z</code>. The intersection of that line <code>d</code> with the horizon generates the vanishing point <code>F2</code> for the projection of the halfline <code>b</code> to the mirror plane. 
* The line between <code>F1</code> and <code>S1</code> is the projective image of the half-line <code>b</code>.


[[File:z_construction_line_on_surface.png|450px|Z-Construction]]

== X-Construction for Point on a Surface ==

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

[[File:x_construction_point_on_surface.png|450px|X-Construction]]

== I-Construction for a Vertical Line ==

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

[[File:i_construction_point_over_surface.png|450px|I-Construction]]

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

[[File:3d_object_perspective_drawing.png|450px|3D-Object]]

== Stereoscopy ==
Spatial perception is required to interact with your environment and relevant for geometric and spatial learning objectives (e.g. in chemistry<ref>Coleman, S. L., & Gotch, A. J. (1998). Spatial perception skills of chemistry students. Journal of Chemical Education, 75(2), 206.</ref>. The constructions mentioned above can be extended to [[Stereoscopy]].


=== Learning Activity - Projection of Circle ===
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 <math>t\in [0, 2\pi]</math> in the plane of the stand on circular path.

=== Animation for the Projection of Circles ===

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.

[[File:Perspektifische Spiegelung eines Kreises mittels Geogebra.gif|350px|the GIF animation shows the projection of a circle]]

=== Learning Activities - Stereoscopy ===
[[File:Tree trunk stereo.png|thumb|Stereoscopic images visualized with two images for left and right eye]]
* 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 ===
The following animation shows the difference of the projections according to the position of left eye <math>Z_L</math> and right eye <math>Z_R</math>.
 
[[File:I-construction stereoscopy animation.gif|450px|I-construction stereoscopy - animation]]

== External Resources ==
* [https://github.com/niebert/3D_Construction_Geogebra/archive/master.zip GitHub Resources with Geogebra Files as ZIP-File]
* [https://github.com/niebert/3D_Construction_Geogebra/ GitHub Repository for Wikiversity Learning Resource]
* [https://www.geogebra.org/m/nzcnrsdz Orthogonal Projection in Schools with Geogebra - by Anna Hundertmark]

== See also ==
* [[3D Modelling]]
* [[3D Modelling/Projective Geometry|Projective Geometry]]
* [[Photogrammetry/Regard3D|Regard3D]]
* [[Photogrammetry]]
* [[w:en:Perspective_(graphical)|Perspective Drawing]]
* [[Equirectangular projection]]
* [[Stereoscopy]]

== References ==
{{Reflist}}

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