Hyper-plane in logistic regression vs linear regression for same number of features

Geometric interpretation of Logistic Regression and Linear regression is considered here.

I was going through Logistic regression and Linear regression. In the optimization equation of both following term is used. $$W^{T}.X$$

W is a vector which holds weights of the hyper-plane.

I realized following about the dimensions of the fitted hyper-plane. Want to confirm it.

Let, d = Number of features for both Logistic Regression and Linear Regression.

Logistic Regression case: Fitted hyper-plane is d-dimensional.

Linear Regression case: Fitted hyper-plane is (d + 1) dimensions.

Example

d = 2

feature 1 : weight, feature 2 : height

Logistic Regression: Its a 2 class classification. y : {obsess, normal)

Linear Regression: y: blood pressure (real value)

Here,

• Logistic Regression will fit a 2-D line.
• Linear Regression will fit a 3-D plane.

Please confirm if this understanding is correct and same happens even in higher dimensions.