Alpinewerk
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Navigating 2D grids with complex numbers.

Grid navigation involves manipulating row and column indices. However, we can take a creative leap by representing grid positions as complex numbers, where the real part corresponds to the row and the imaginary part to the column. This approach simplifies grid traversal and allows for concise and intriguing operations, leveraging the mathematical properties of complex numbers.

GRID = [
    ["NW", "N", "NE"],
    ["W",  "C",  "E"],
    ["SW", "S", "SE"],
]

# Given a grid, dynamically define the starting point, the center.
# In our case, this would be the "C", at [1, 1].
row_center = len(GRID) // 2     # 1.
col_center = len(GRID[0]) // 2  # 1.


# We define the starting point as a complex number (1+1j), which is a number
# with a real and imaginary part. The Real part represents the row and the
# Imaginary part represents the column.
starting_point = complex(row_center, col_center)
print(starting_point)  # (1+1j)
#                         │└┬┘
#                         │ │
#                         │ └── Imaginary part, denoted by the "j" suffix.
#                         └── Real part.


# We can move around the grid by adding or subtracting either a real and/or
# imaginary part. Note, if either part is 0, we can omit it. We can re-imagine
# the grid with "C" as the starting point as following:

# (-1-1j) │ (-1+0j) │ (-1+1j)
# ────────────────────────────
# (0-1j)  │   "C"   │ (0+1j)
# ────────────────────────────
# (+1-1j) │ (+1+0j) │ (+1+1j)

# From "C" to move down towards "S", we add 1:
starting_point + 1

# From "C" to move right towards "E", we subtract 1j:
starting_point - 1j

# From "C" to move diagonally towards "NW", we subtract -1-1j:
starting_point - 1 - 1j

# These operations result in another complex number, which we can then break
# down into a row and column to get the value of the grid. E.g.
position = starting_point + 1         # -> "S".
position += 1j                        # -> "SE".
position += -2                        # -> "NE".
position += 1 - 2j                    # -> "W".

# These operations can be chained together. E.g.
starting_point + 1 + 1j - 2 + 1 - 2j  # -> "W".

# To break down the complex number into the row and column, we can simply
# extract the parts:
row = int(position.real)
col = int(position.imag)
print(GRID[row][col])  # "W".

Alpinewerk Digital e.U. Owner: Alexander Semerad

registered sole proprietorship

Steckhovengasse 18, 1130 Wien - AT

Tel: +43 (0) 681 20289940

E-Mail: [email protected]

VAT-Nr: ATU75303217

FN: 544875m

Commercial Court Vienna

I. Overview

Welcome to the website www.alpinewerk.com! Please get a picture of how we process your personal data when you visit our website or you are otherwise in a business relationship with us (Art 13, Art 14 GDPR; § 96 Abs 3 TKG – the Austrian Telecommunication Act).

II. What data do we process when you visit our website?

The following data may be processed when you visit our website:

  • Browser type,
  • operating system,
  • country,
  • date,
  • time and duration of access,
  • IP address and pages visited on our website including entry and exit pages,
  • data you enter via a contact form,

The processing of this data is necessary to manage the security of the operation of the website and to ensure the functionality of the website from a technical point of view. The processing of this data is justified by our legitimate interest in the operation of our website (Art 6 para 1 lit f DSGVO).

For the operation of our website, it may be necessary for us to disclose your data to the following recipients:

Recipients of data Purpose of data processing Reasoning for data processing Location Third party transfer
Alphabet Inc (Google Analytics) Statistical evaluations and marketing purposes Consent (Art 6 Abs 1 lit a GDPR) USA Standard contractual clauses (Art 46 Abs 2 lit c DSGVO)

II.1. Overview of the „technical“ cookies used

The above-mentioned data is stored via so-called "cookies". These technical cookies are only used to the extent necessary (§ 96 para 3 TKG). Cookies are text files that are stored on your computer and enable an analysis of the use of the website. They are used to recognise and store temporary data of the homepage visitor. We only use cookies to the extent necessary to communicate with you via the homepage.

These technical cookies are activated as soon as you visit our homepage.

The following cookies are used on our homepage on the basis of our legitimate interest (Art 6 para 1 lit f GDPR):

Name of the cookie Purpose of the Cookies Storage duration Location of recipient
SESS# Retains the user's states for all page Session USA

II.2. Overview of the „advertising cookies“ used

In addition to the "technical cookies" described above, we also use so-called advertising cookies (including "statistical cookies"). These advertising cookies enable us to better understand and evaluate your interests. With the help of the advertising cookies, we can merge your "surfing behaviour" across the boundaries of our website with data from other websites. This is to enable us to better understand the interests of our homepage visitors and to target them more effectively.

We respect that not every visitor to the website wants this. Therefore, we only process your data in the course of advertising cookies if you consent to this (Art 6 para 1 lit a GDPR). You can revoke this consent at any time, whereby the data processing carried out up to the time of revocation remains justified.

These advertising cookies will only be activated once you have consented to them.

Name of the cookie Purpose of the Cookies Storage duration Location of recipient Purpose of data transfer
_ga (Google Analytics) Statistical purposes 2 years USA Registers a unique ID that is used to generate statistical data on how the visitor uses the website.
ga# Statistical purposes 2 years USA Collects data on how often a user visits a website, as well as data for the first and last visit.

III. For what purposes do we process your data when we are in a business rela-tionship?

In the course of our business relationship with customers or suppliers, we process your data on the basis of contractual (processing of the contractual relationship with you, pre-contractual obli-gations, invoicing of services, dispatch of documents, communication for the processing of the contract) and legal obligations (legally required storage within the meaning of § 132 BAO) (Art 6 para 1 lit b and c GDPR) as well as on the basis of our legitimate interests or on the basis of the legitimate interests of third parties (Art 6 para 1 lit f GDPR), namely:

  • for the purpose of internal administration and management of your business case to the extent necessary (e.g.: Processing your business case, forwarding your business case to various de-partments, filing, archiving purposes, correspondence with you);

  • for the purpose of direct advertising (e.g.: Mailing, e-mailing, satisfaction surveys, congratulatory letters, statistical evaluations);

We would like to expressly inform you that you can object to the processing of your data for the purpose of direct advertising.

  • assertion and defence of legal claims

in each case to the extent necessary. The processing of your data serves the purpose of initiating, maintaining and handling our business relations. If you do not provide us with this data, we will not be able to process your business case.

If particularly sensitive data is processed in the course of specific legal disputes, this is done for the purpose of asserting legal claims (Art 9 (2) (f) in conjunction with Art 6 (1) (b) GDPR).

IV. How long do we store your data?

We will only store your data for as long as is necessary for the purposes for which we collected your data. In this context, statutory retention obligations must be taken into account (for example, for reasons of tax law, contracts and other documents from our contractual relationship must generally be retained for a period of seven years (§ 132 BAO)). In justified individual cases, such as for the assertion and defence of legal claims, we may also store your data for up to 30 years after termination of the business relationship.

We store data of interested parties for up to three years from the time of the last contact by the respective person.

V. Who maybe receives your data?

In the course of our business relationship, it may be necessary for us to transmit your data to the following recipients:

Recipient Purposes Legal justification Location (country) Purpose of data transfer
Bank (credit institution) Payment-services Contractual obligation (Art 6 Abs 1 lit b
GDPR) Austria Within the EAA
Business and tax advisors Tax advisory Contractual obligation (Art 6 Abs 1 lit b
GDPR) Austria Within the EAA
Attorney at law Enforcement of claim Contractual obligation (Art 6 Abs 1 lit b
GDPR) Austria Within the EAA

VI. Collection of data from other sources (Art 14 GDPR)

In the course of a business relationship or the initiation thereof, it is naturally necessary to conduct research on the business partner. This is done exclusively to the extent necessary for this purpose. In this context, data may be retrieved and processed from the following sources:

Source Publicly available Data subjects Purpose/Reasoning
Internet-Research Ja Contact details Contact for business purposes

VII. Automated decision-making and profiling (Art 13 Abs 2 lit f GDPR)?

No automated decision-making or profiling takes place in our company.

VIII. What are your rights regarding the processing of your data?

We would like to inform you that, provided the legal requirements are met, you have the right to request information about your personal data:

  • You have the right to request information about which of your data is processed by us (see in detail Art 15 of the GDPR).
  • You have the right to request the correction or completion of incorrect or incomplete data relating to you (see in detail Art 16 of the GDPR).
  • Have the right to have your data deleted (see in detail Art 17 GDPR).
  • Have the right to object to processing of your data which is necessary to protect our legitimate interests or those of a third party. This applies in particular to the processing of your data for advertising purposes.
  • Have the right to obtain the transfer of the data you have provided in a structured, commonly used and machine-readable format.

If we process your data on the basis of your consent, you have the right to revoke this consent at any time by email. This does not affect the lawfulness of the data processing carried out up to this point (Art 7 (3) DSGVO).

IX. Do you have a right to complain?

If, contrary to expectations, there is a breach of your right to lawful processing of your data, please contact us by post or e-mail. We will endeavour to deal with your concern promptly. However, you also have the right to lodge a complaint with the supervisory authority for data protection matters responsible for you.

The address of the Austrian data protection authority is:

Austrian Data Protection Authority Barichgasse 40-42, 1030 Vienna

X. How can you get in contact with us?

If you have any further questions about the processing of your data, please do not hesitate to contact our data protection coordinator using the contact details below.

XI. Controller

Controller pursuant to Art 4 Z 7 GDPR:

Alpinewerk Digital e.U. Alexander Semerad

Steckhovengasse 18, 1130 Wien - AT

[email protected] +4368120289940

FN 544875m

Author: Attorney Dr. Tobias Tretzmüller, LL.M (IT-LAW); https://www.digital-recht.at/