Ευκλείδειον Επίπεδον
Είναι ένα επίπεδο.
Ετυμολογία[]
Η ονομασία "Ευκλείδειο" σχετίζεται ετυμολογικά με το όνομα "Ευκλέιδης".
Εισαγωγή[]
Απόσταση[]
The Euclidean distance between two points of the plane with Cartesian coordinates and is
This is the Cartesian version of Pythagoras' theorem. In three-dimensional space, the distance between points and is
which can be obtained by two consecutive applications of Pythagoras' theorem.
Ευκλείδειοι Μετασχηματισμοί=[]
Μεταφορά (Translation)[]
translation a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (X,Y) to the Cartesian coordinates of every point in the set.
That is, if the original coordinates of a point are (x,y), after the translation they will be
Κλιμάκωση (Scaling)=[]
To make a figure larger (μεγέθυνση) or smaller (Σμίκρυνση) is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If (x,y) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates
If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller.
Περιστροφή (Rotation)=[]
To rotate a figure counterclockwise around the origin by some angle is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x',y'), where
Thus:
Κατοπτρισμός (Reflection)=[]
If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the X axis).
Γενικός Μετασχηματισμός (General transformations)=[]
The Euclidean transformations of the plane are the translations, rotations, scalings, reflections, and arbitrary compositions thereof. The result of applying a Euclidean transformation to a point is given by the formula
where A is a 2×2 matrix and b is a pair of numbers, that depend on the transformation; that is,
The matrix A must have orthogonal rows with same Euclidean length, that is,
and
This is equivalent to saying that A times its transpose must be a diagonal matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane.
The formulas define a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that
Μερικοί Μετασχηματισμοί[]
In two-dimensional space R2 linear maps are described by 2 × 2 real matrices. These are some examples:
- rotation by 90 degrees counterclockwise:
- rotation by θ degrees counterclockwise:
- reflection against the x axis:
- reflection against the y axis:
- scaling by 2 in all directions:
- horizontal shear mapping:
- squeeze mapping:
- projection onto the y axis:
Υποσημειώσεις[]
Εσωτερική Αρθρογραφία[]
- Ευκλείδειος Χώρος
- Επίπεδο
- [[]]
Βιβλιογραφία[]
Ιστογραφία[]
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Αν και θα βρείτε εξακριβωμένες πληροφορίες "Οι πληροφορίες αυτές μπορεί πρόσφατα Πρέπει να λάβετε υπ' όψη ότι Επίσης, |
- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν
- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)