projizieren
to project
To cast an image, a film, or light onto a surface using a projector.
Für den Vortrag projizieren wir die Folien an die weiße Wand.
For the presentation, we will project the slides onto the white wall.

to project
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to project
To cast an image, a film, or light onto a surface using a projector.
Für den Vortrag projizieren wir die Folien an die weiße Wand.
For the presentation, we will project the slides onto the white wall.

The word comes from the Latin 'proicere' (to throw forward). This basic idea appears in all meanings: you 'throw' an image onto the wall, 'throw' your own feelings onto others, or 'throw' a 3D shape onto a 2D surface.
Er projiziert seine eigenen Ängste auf seine Kinder.
He projects his own fears onto his children.
All meanings often follow the same pattern: 'etwas (Akkusativ) auf etwas/jemanden (Akkusativ) projizieren' (to project something onto something/someone). The preposition 'auf' (onto) is therefore central to the sentence construction.
Der Lehrer projiziert das Arbeitsblatt an die Tafel.
The teacher projects the worksheet onto the board.
The context reveals the meaning: if it is about technology (projector, film), the first meaning is intended. If it is about feelings and psychology, it is the second. If it is about mathematics or geography, it is the third.
Im Kino wird der Film auf eine riesige Leinwand projiziert.
In the movie theater, the film is projected onto a huge screen.
The psychological meaning of 'projizieren' (to project) is technical language, but it is also used in educated everyday language to describe transferring one's own feelings.
Im Streit projizierte er seine Unzufriedenheit auf seine Partnerin.
During the argument, he projected his dissatisfaction onto his partner.
to throw
'werfen' and 'projizieren' can both have to do with light or an image on a surface. Here, 'werfen' is more general and more natural with shadows: a tree 'wirft einen Schatten' (casts a shadow). 'projizieren' is used when an image, a film, or light is put onto a wall or screen with a device. Learners may confuse the words because you can also say that a projector 'wirft' an image onto a wall. However, 'projizieren' sounds precise and technical.

Für die Präsentation projiziert Nele die Fotos an die weiße Wand.
For the presentation, Nele projects the photos onto the white wall.
Am Nachmittag wirft der Baum einen langen Schatten auf den Weg.
In the afternoon, the tree casts a long shadow on the path.